A Collection of Sharp Dilation Invariant Inequalities for Differentiable Functions

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2 2

ai,j uxi uxj dx C ∆u dx , R2 ≤ R2 | | Z i,j=1 Z X ∞ R2 where ai,j = const and u is an arbitrary complex-valued function in C0 ( ), may hold if and only if a11 +a22 = 0. We also ﬁnd the best constant C in the last inequality. This is a particular case of Theorem 1 proved in Section 1. In Section 2 we establish a new weighted G˚arding type inequality

2 2 −1 2 −1 2u log(e x ) dx Re ∆ u u log x dx (3) R2 |∇ | | | ≤ R2 · | | Z Z ∞ R2 for all u C0 ( 0 ). Estimates of such a kind proved to be useful in the study of boundary∈ behavior\{ of} solutions to elliptic equations (see [M4], [MT], [M6], [M7], [E], [MM]). Before turning to the contents of the next section, we introduce some notation. Rn Rn Rn−1 By + we denote the half-space x = (x1,...xn) , xn > 0 . Also let = Rn ∞ Rn ∞ {Rn ∈ } ∂ +. As usual, C0 ( +) and C0 ( +) stand for the spaces of inﬁnitely diﬀerentiable Rn Rn functions with compact support in + and +, respectively. In Section 3 we are concerned with the inequality

2 2 u ∞ Rn xn u dx Λ 2 | | 2 1/2 dx, u C0 ( +). (4) Rn |∇ | ≥ Rn (x + x ) ∈ Z + Z + n−1 n It was obtained in 1972 by one of the authors and proved to be useful in the study of the generic case of degeneration in the oblique derivative problem for second order elliptic diﬀerential operators [M2]. −1/2 By substituting u(x)= xn v(x) into (4), one deduces with the same Λ that

2 2 2 1 v dx v dx v dx | | 2 +Λ 2| | 2 1/2 (5) Rn |∇ | ≥ 4 Rn xn Rn xn(x + x ) Z + Z + Z + n−1 n for all v C∞(Rn ) (see [M5], Sect. 2.1.6). ∈ 0 + Another inequality of a similar nature obtained in [M5] is

2 2 1 v γ 2 v dx | | dx + C x v Rn . (6) 2 n Lq( +) Rn |∇ | ≥ 4 Rn xn k k Z + Z + (This is a special case of inequality (2.1.6/3) in [M5].) Without the second term in the right-hand sides of (5) and (6), these inequalities reduce to the classical Hardy inequality with the sharp constant 1/4 (see [Da]). An interesting feature of (5) and (6) is their dilation invariance. Variants, extensions, and reﬁnements of (5) and (6), usually called Hardy’s inequalities with remainder term, became the theme of many subsequent studies ([ACR], [Ad], [AGS], [BCT], [BFT1], [BFT2], [BFL], [BM], [BV], [CM], [DD], [DNY], [EL],

2 [FMT1]–[FMT3], [FT], [FTT], [FS], [GGM], [HHL], [TT], [TZ], [Ti1], [Ti2], [VZ], [YZ] et al). In Theorem 3, proved in Section 3, we ﬁnd a condition on the function q which is necessary and suﬃcient for the inequality

2 2 2 1 v dx xn v dx v dx | | 2 C q 2 2 1/2 2| | 2 1/2 , (7) Rn |∇ | − 4 Rn xn ≥ Rn (xn−1 + xn) xn (xn−1 + xn) Z + Z + Z + ∞ Rn where v is an arbitrary function in C0 ( +). This condition implies, in particular, that the right-hand side of (5) can be replaced by

v 2dx C | | 2 . Rn xn + 2 Z xn 1 log 2 2 1/2 − (xn−1 + xn) The value Λ = 1/16 in (4) obtained in [M2] is not best possible. Tidblom replaced it by 1/8 in [Ti2]. As a corollary of Theorem 3, we ﬁnd an expression for the optimal value of Λ.

Let the measure µb be deﬁned by dx µ (K)= (8) b x b ZK | | for any compact set K in Rn. In Section 4 we obtain the best constant in the inequality

1/p p dx u Lτ,q(µb) C u(x) a , k k ≤ Rn |∇ | x Z | | where the left-hand side is the quasi-norm in the Lorentz space (µ ), i.e. Lτ,q b ∞ q/τ 1/q u = µ x : u(x) t d(tq ) . k kLτ,q(µb) b{ | |≥ } Z0 As a particular case of this result we obtain the best constant in the Hardy-Sobolev inequality 1/q 1/p q dx p dx u(x) b u(x) a , (9) Rn | | x ≤C Rn |∇ | x Z | | Z | | ﬁrst proved by Il’in in 1961 in [Il] (Th. 1.4) without discussion of the value of . Our result is a direct consequence of the capacitary integral inequality from [M7] combinedC with an isocapacitary inequality. For particular cases the best constant was found in [CW] (p = 2), in [M3], Sect. 2 (p = 1, a = 0), in [GMGT] (p = 2, n =C 3, a = 0), in [L] (p = 2, n 3, a = 0), and in [Na] (1

3 1 Estimate for a quadratic form of the gradient

n Theorem 1 Let n 2 and let A = ai,j i,j=1 be an arbitrary matrix with constant complex entries. The≥ inequality k k

n+2 2 A u, u Cn dx C ( ∆) 4 u dx , (10) Rn h ∇ ∇ i ≤ Rn − Z Z ∞ Rn where C is a positive constant, holds for all complex-valued u C0 ( ) if and only if the trace of A is equal to zero. The best value of C is given∈ by

(4π)−n/2 C = max ai,j ωiωj , (11) Γ n +1 ω∈Sn−1 2 1≤i,j≤n X where Sn−1 is the (n 1)-dimensional unit sphere in Rn. − (The notation ( ∆)s in (10) stands for an integer or noninteger power of ∆.) − − Proof. By we denote the unitary Fourier transform in Rn deﬁned by F h(ξ)=(2π)−n/2 h(x) e−ix·ξ dx. (12) F Rn Z We set h = ( ∆)(n+2)/4u and write (10) in the form − 2 2 ξ ξ dξ h(ξ) A , n C h(x) dx . (13) Rn |F | ξ ξ Cn ξ ≤ Rn | | Z D | | | |E | | Z

The singular integral in the left-hand side exists in the sense of the Cauchy principal value, since −1 n−1 Aω,ω Cn dsω = n S Tr A =0, n−1 h i | | ZS where Tr A is the trace of A (see, for example, [MP], Ch. 9, Sect. 1 or [SW], Theorem 4.7). Let ξ ξ k(ξ)= ξ −n A , . | | ξ ξ Cn D | | | | E The left-hand side in (13) equals

−1 k(ξ) h (ξ) (x) h(x) dx = (2π)−n/2 ( −1k) h (x) h(x) dx Rn F F Rn F ∗ Z Z with meaning the convolution. Thus, inequality (13) becomes ∗ 2 ( −1k) h (x) h(x) dx (2π)n/2 C h(x) dx . (14) Rn F ∗ ≤ Rn | | Z Z We note that for ξ Rn, ∈ n n k(ξ)= ξ −n−2 a ξ2 n−1 ξ 2 + a ξ ξ . (15) | | jj j − | | ij i j j=1 i,j=1 X Xi=6 j Hence, for n> 2 n 1 ∂2 k(ξ)= a ξ 2−n, (16) n (n 2) ij ∂ξ ∂ξ | | i,j=1 i j − X

4 and for n =2 2 1 ∂2 k(ξ)= a log ξ −1. (17) 2 ij ∂ξ ∂ξ | | i,j=1 i j X Applying −1 to the identity F ∂2 ξ 2−n ∂2 ∆ξ n−|1| = δ(ξ), − ∂ξi∂ξj S (n 2) ∂ξi∂ξj | | − where n> 2 and δ is the Dirac function, we obtain from (16)

Sn−1 n x x −1k (x)= −| | a i j . (18) F n (2π)n/2 ij x 2 i,j=1 X | | Here Sn−1 stands for the (n 1)-dimensional measure of Sn−1: | | − n n−1 2π 2 S = n . (19) | | Γ( 2 ) Hence 2−n/2 n x x −1k (x)= − a i j . (20) F Γ(1 + n ) ij x 2 2 i,j=1 X | | Similarly, we deduce from (17) that (20) holds for n = 2 as well. Now, (10) with C given by (11) follows from (20) inserted into (14). Next, we show the sharpness of C given by (11). Let θ denote a point on Sn−1 such that −1k (θ) = max −1k(ξ) . (21) F ξ∈Rn\{0} |F | In order to obtain the required lower estimate for C, it suﬃces to set x h(x)= η( x ) δ , | | θ x | | ∞ n−1 where η C0 [0, ), η 0, and δθ is the Dirac measure on S concentrated at θ, into the∈ inequality∞ (14).≥ (The legitimacy of this choice of h can be easily checked by approximation.) Then estimate (14) becomes ∞ ∞ ρ r −1k − θ η(r) rn−1 η(ρ) ρn−1drdρ 0 0 F ρ r Z Z | − | ∞ 2 (2π)n/2 C η(ρ) ρn−1dρ . (22) ≤ 0 Z In view of (18) and (20),

−1k ( θ)= −1k (θ) F ± F which together with (21) enables one to write (22) in the form

max −1k(ξ) (2π)n/2 C. ξ∈Rn\{0} |F |≤ By (18) and (20) this can be written as

n−1 S n/2 | | max aij ωiωj (2π) C. n (2π)n/2 ω∈Sn−1 ≤ 1≤i,j≤n X

5 The result follows from (19).

Remark 1 Let P and Q be functions, positively homogeneous of degrees 2m and m + n/2 respectively, m > n/2. We assume that the restrictions of P , Q, and P Q −2 to Sn−1 belong to L−1(Sn−1), By the same argument as in Theorem 1, one concludes| | that the condition P (ω) 2 dsω = 0 (23) n−1 Q(ω) ZS | | is necessary and suﬃcient for the inequality

2 P (D)u u dx C Q(D) u dx (24) Rn · ≤ Rn Z Z ∞ Rn to hold for all u C 0 ( ). Moreover, using the classical formula for the Fourier transform of a positively∈ homogeneous function of degree n (see Theorem 4.11 in [SW]1), one obtains that the best value of C in (24) is given− by

iπ P (θ) sup sgn(θ ω) + log θ ω 2 dsθ . (25) ω∈Sn−1 Sn−1 2 · | · | Q(θ) Z | |

In particular, if P (ω)/ Q(ω) 2 is a spherical harmonic, the best value of C in (24) is equal to | | (4π)−n/2Γ(m) P (ω) n max | | , (26) Γ + m ω∈Sn−1 Q(ω) 2 2 | | which coincides with (11) for m = 1, P(ξ)= Aξ ξ, and Q(ξ)= ξ 1+n/2. · | | 2 Weighted G˚arding inequality for the biharmonic operator

We start with an auxiliary Hardy type inequality.

Lemma 1 Let u C∞(R2). Then the sharp inequality ∈ 0

dx 2 Re x1ux1 + x2ux2 ∆u 2 ∆u dx (27) R2 x ≤ R2 | | Z | | Z holds.

Proof. Let (r, ϕ) denote polar coordinates in R2 and let

∞ ikϕ u(r, ϕ)= uk(r)e . k=X−∞ Then (27) is equivalent to the sequence of inequalities

∞ 2 ∞ 2 2 ′′ 1 ′ k ′ ′′ 1 ′ k Re v + v 2 v v dr v + v 2 v r dr, k =0, 1, 2,... (28) 0 r − r ≤ 0 r − r Z Z 1 Note that the deﬁnition of the Fourier transform in [SW] contains exp(−2πix · ξ) unlike (12).

6 ∞ −1 where v is an arbitrary function on C0 ([0, )). Putting t = log r and w(t) = v(e−t), we write (28) in the form ∞

2 Re w′′ k2w w′ e2t dt w′′ k2w e2t dt R1 − ≤ R1 − Z Z which is equivalent to the inequality

2 Re g′′ 2g′ + (1 k2)g g′ g dt g′′ 2g′ + (1 k2)g dt, (29) R1 − − − ≤ R1 − − Z Z where g = etw. Making use of the Fourier transform in t, we see that (29) holds if and only if for all λ R1 and k =0, 1, 2 ... ∈ 2 Re λ2 +1 k2 2iλ 1 iλ λ2 1+ k2 +4λ2, − − − − ≤ −

which is the same as 3x 1+ k2 x2 + 2(k2 + 1)x + (k2 1)2 − ≤ − 2 with x = λ . This elementary inequality becomes equality if and only if k = 0 and x = 0.

Remark 2 In spite of the simplicity of its proof, inequality (27) deserves some interest. Let us denote the integral over R2 in the left-hand side of (27) by Q(u,u) and write (27) as Re Q(u,u) ∆u 2 R2 . | |≤k kL ( ) However, the absolute value of the corresponding sesquilinear form Q(u, v) cannot be majorized by C ∆u L2(R2) ∆v L2(R2). Indeed, the opposite assertion would yield an k k−1 k k 2 R2 upper estimate of r ∂u/∂r L2(R2) by the norm of ∆u in L ( ) which is wrong for a function linear neark the origin.k

Remark 3 Note that under the additional orthogonality assumption

2π u(r, ϕ)dϕ =0 for r> 0 (30) Z0 the above proof of Lemma 1 provides inequality (27) with the sharp constant factor 3/4 in the right-hand side. Besides, (30) implies

dx Re x1ux1 + x2ux2 ∆u 2 0. R2 x ≤ Z | |

Using (27), we establish a new weighted G˚arding type inequality.

Theorem 2 Let u C∞(R2 0 ). Then inequality (3) holds. ∈ 0 \{ } Proof. Clearly, the right-hand side in (3) is equal to

Re ∆u ∆ (u log x −1) dx R2 | | Z = ∆u 2 log x −1dx + 2Re ∆u u log x −1dx. R2 | | | | R2 · ∇ · ∇ | | Z Z

7 Combining this identity with (27), we arrive at the inequality

∆u 2 log(e2 x )−1dx Re ∆u ∆(u log x −1) dx. (31) R2 | | | | ≤ R2 | | Z Z Note that ∆u ∆u log x −1 = u (∆u log x −1) dx R2 · · | | − R2 ∇ · ∇ · | | Z Z 2 ∂u ∂ ∂ ∂ = Re u log x −1 + u u log x −1 dx, R2 ∇∂x · ∂x ∇ · | | ∇ · ∂x ∇ · ∂x | | Z j=1 j j j j X which is equal to

2 2 ∂2u 2 1 ∂ ∂ log x −1 + u 2 log x −1 dx. R2 ∂xj ∂xk | | 2 ∂xj |∇ | · ∂xj | | ! Z j,k=1 j=1 X X Integrating by parts in the second term, we see that it vanishes. Thus, we conclude that 2 −1 2 −1 ∆u log x dx = 2u log x dx R2 | | | | R2 |∇ | | | Z Z which together with (27) and the obvious identity

2 2 ∆u dx = 2u dx R2 | | R2 |∇ | Z Z completes the proof of (3). In order to see that no constant less than 1 is admissible in front of the integral in the right-hand side of (3), it suﬃces to put u(x)= eihx,ξi η(x) with η C∞(R2) 0 into (3) and take the limit as ξ . ∈ 0 \{ } | | → ∞ Remark 4 If the condition u = 0 near the origin in Theorem 2 is removed, the above proof gives the additional term

π u(0) 2 2 Re u(0) ∆u(0) |∇ | − in the right-hand side of (3).

3 Dilation invariant Hardy’s inequalities with remainder term

Theorem 3 (i) Let q denote a locally integrable nonnegative function on (0, 1). The best constant in the inequality 2 2 xn u xn u dx C q 2 2 1/2 2 | | 2 1/2 dx, (32) Rn |∇ | ≥ Rn (xn−1 + xn) (xn−1 + xn) Z + Z + for all u C∞(Rn ), which is equivalent to (7), is given by ∈ 0 + π/2 2 1 2 y′(ϕ) + y(ϕ) sin ϕ dϕ 0 4 λ := inf Z π/2 , (33) 2 y (ϕ) q (sin ϕ ) dϕ Z0

8 where the inﬁmum is taken over all smooth functions on [0,π/2]. (ii) Inequalities (32) and (7) with a positive C hold if and only if

t sup (1 log t) q(τ) dτ < . (34) − ∞ t∈(0,1) Z0 Moreover, t −1 λ sup (1 log t) q(τ) dτ , (35) ∼ t∈(0,1) − 0 Z where a b means that c a b c a with absolute positive constants c and c . ∼ 1 ≤ ≤ 2 1 2 ∞ R2 ∞ Rn−2 ′ Proof (i) Let U C0 ( +), ζ C0 ( ), x = (x1,...xn−2), and let N = const > 0. Putting ∈ ∈ (2−n)/2 −1 ′ u(x)= N ζ(N x )U(xn−1, xn) into (32) and passing to the limit as N , we see that (32) is equivalent to the inequality → ∞

2 2 2 x2 U dx1dx2 x2 Ux1 + Ux2 dx1dx2 C q 2 2 1/2 | 2| 2 1/2 , (36) R2 | | | | ≥ R2 (x + x ) (x + x ) Z + Z + 1 2 1 2 ∞ R2 R2 where U C0 ( +). Let (ρ, ϕ) be the polar coordinates of the point (x1, x2) +. Then (36)∈ can be written as ∈

∞ π ∞ π U 2 + ρ−2 U 2 sin ϕdϕρ2dρ C U 2q(sin ϕ) dϕdρ. | ρ| | ϕ| ≥ | | Z0 Z0 Z0 Z0 By the substitution U(ρ, ϕ)= ρ−1/2v(ρ, ϕ) the left-hand side becomes ∞ π π ∞ 2 2 1 2 dρ ρvρ + vϕ + v sin ϕ dϕ Re v vρ dρ sin ϕ dϕ. (37) 0 0 | | | | 4| | ρ − 0 0 Z Z Z Z Since v(0) = 0, the second term in (37) vanishes. Therefore, (36) can be written in the form ∞ π ∞ π 2 2 1 2 dρ 2 dρ ρvρ + vϕ + v sin ϕ dϕ C v q(sin ϕ) dϕ . (38) 0 0 | | | | 4| | ρ ≥ 0 0 | | ρ Z Z Z Z Now, the deﬁnition (33) of λ shows that (38) holds with C = λ. In order to show the optimality of this value of C, put t = log ρ and v(ρ, ϕ) = w(t, ϕ). Then (38) is equivalent to

π π 2 2 1 2 2 wt + wϕ + w sin ϕ dϕ dt C w q(sin ϕ)dϕ dt. (39) R1 | | | | 4| | ≥ R1 | | Z Z0 Z Z0 Applying the Fourier transform w(t, ϕ) wˆ(s, ϕ), we obtain → π π 2 2 1 2 2 wˆϕ + s + wˆ sin ϕ dϕ ds C wˆ q(sin ϕ)dϕ ds. (40) R1 0 | | | | 4 | | ≥ R1 0 | | Z Z Z Z Putting here wˆ(s, ϕ)= ε−1/2η(s/ε)y(ϕ),

9 ∞ R1 ∞ where η C0 ( ), η L2(R1) = 1, and y is a function on C ([0, π]), and passing to the limit∈ as ε 0, wek arrivek at the estimate → π 1 π y′(ϕ) 2 + y(ϕ) 2 sin ϕ dϕ C y(ϕ) 2q(sin ϕ)dϕ (41) 0 | | 4| | ≥ 0 | | Z Z where π can be changed for π/2 by symmetry. This together with (33) implies Λ λ. The proof of (i) is complete. ≤ ϕ (ii) Introducing the new variable ξ = log cot 2 , we write (33) as

∞ 2 ′ 2 z(ξ) z (ξ) + | | 2 dξ 0 | | 4 (cosh ξ) λ = inf Z ∞ . (42) z 1 dξ z(ξ) 2 q 0 | | cosh ξ cosh ξ Z Since 1 z(0) 2 2 z′(ξ) 2 + z(ξ) 2 dξ | | ≤ | | | | Z0 and ∞ e2ξ ∞ dξ ∞ e2ξ z(ξ) 2 dξ 2 z(ξ) z(0) 2 +2 z(0) 2 dξ | | (1 + e2ξ)2 ≤ | − | ξ2 | | (1 + e2ξ)2 Z0 Z0 Z0 ∞ 8 z′(ξ) 2dξ + z(0) 2, ≤ | | | | Z0 it follows from (42) that

∞ z′(ξ) 2dξ + z(0) 2 0 | | | | λ inf ∞Z . (43) ∼ z 1 dξ z(ξ) 2 q 0 | | cosh ξ cosh ξ Z Setting z(ξ) = 1 and z(ξ) = min η−1ξ, 1 for all positive ξ and ﬁxed η > 0 into the ratio of quadratic forms in (43), we{ deduce} that

∞ 1 dξ −1 ∞ 1 dξ −1 λ min q , sup η q . ≤ 0 cosh ξ cosh ξ η>0 η cosh ξ cosh ξ nZ Z o Hence, t −1 λ c sup (1 log t) q(τ) dτ . ≤ t∈(0,1) − 0 Z In order to obtain the converse estimate, note that

∞ 1 dξ ∞ 8 sup η q z′(ξ) 2dξ η>0 η cosh ξ cosh ξ 0 | | Z Z

10 (see, for example, [M5], Sect. 1.3.1). Therefore,

∞ 1 dξ ∞ 1 dξ z(ξ) 2 q 8max q , 0 | | cosh ξ cosh ξ ≤ ( 0 cosh ξ cosh ξ Z Z ∞ 1 dσ ∞ sup η q z′(ξ) 2dξ + z(0) 2 η>0 η cosh σ cosh σ ) 0 | | | | Z Z which together with (43) leads to the lower estimate

∞ 1 dξ −1 ∞ 1 dξ −1 λ min q , sup η q . ≥ 0 cosh ξ cosh ξ η>0 η cosh ξ cosh ξ nZ Z o Hence, t −1 λ c sup (1 log t) q(τ) dτ . ≥ t∈(0,1) − 0 Z The proof of (ii) is complete. Since (34) holds for q(t)= t−1(1 log t)−2, Theorem 3 (ii) leads to the following assertion. −

Corollary 1 There exists an absolute constant C > 0 such that the inequality

2 2 2 1 v dx v dx v dx | | 2 C | | 2 (44) Rn |∇ | − 4 Rn x ≥ Rn xn + + n + 2 Z Z Z xn 1 log 2 2 1/2 − (xn−1 + xn) holds for all v C∞(Rn ). The best value of C is equal to ∈ 0 + π 2 1 2 y′(ϕ) + y(ϕ) sin ϕ dϕ 0 4 λ := inf πZ , (45) h 2 i y(ϕ ) (sin ϕ)−1 1 log sin ϕ)−2dϕ − Z0 where the inﬁmum is taken over all smooth functions on [0,π/2]. By numerical approximation, λ =0.16 ...

A particular case of Theorem 3 corresponding to q = 1 is the following assertion.

Corollary 2 The sharp value of Λ in (4) and (5) is equal to

π 2 1 2 y′(ϕ) + y(ϕ) sin ϕ dϕ 0 4 λ := inf Z π , (46) h 2 i y(ϕ ) dϕ Z0 where the inﬁmum is taken over all smooth functions on [0, π]. By numerical approximation, λ =0.1564 ...

Remark 5 Let us consider the Friedrichs extension ˜ of the operator L ′ sin ϕ : z (sin ϕ)z′ + z (47) L →− 4

11 deﬁned on smooth functions on [0, π]. It is a simple exercise to show that the energy space of ˜ is compactly imbedded into L2(0, π). Hence, the spectrum of ˜ is discrete and λ deﬁnedL by (46) is the smallest eigenvalue of ˜. L L Remark 6 The argument used in the proof of Theorem 3 (i) with obvious changes enables one to obtain the following more general fact. Let P and Q be measurable nonnegative functions in Rn, positive homogeneous of degrees 2µ and 2µ 2, respectively. The sharp value of C in −

2 2 ∞ Rn P (x) u dx C Q(x) u dx, u C0 ( ), (48) Rn |∇ | ≥ Rn | | ∈ Z Z is equal to 2 n 2 2 P (ω) ωY + µ 1+ Y dsω Sn−1 |∇ | − 2 | | λ := inf Z , 2 Q(ω) Y dsω n−1 | | ZS where the infimum is taken over all smooth functions on the unit sphere Sn−1. A direct consequence of this assertion is the following particular case of (48). Remark 7 Let p and q stand for locally integrable nonnegative functions on (0, 1] and let µ R1. If n> 2, the best value of C in ∈ x x x 2µp n u 2dx C x 2µ−2q n u 2dx, (49) Rn | | x |∇ | ≥ Rn | | x | | Z | | Z | | where u C∞(Rn), is equal to ∈ 0 π n 2 y′(θ) 2 + µ 1+ y(θ) 2 p(cos θ)(sin θ)n−2dθ | | − 2 | | inf Z0 , (50) π y(θ) 2q(cos θ)(sin θ)n−2dθ | | Z0 with the infimum taken over all smooth functions on the interval [0, π]. Formula (50) enables one to obtain a necessary and sufficient condition for the existence of a positive C in (49). Let us assume that the function (sin θ)2−n θ → p(cos θ) is locally integrable on (0, π). We make the change of variable ξ = ξ(θ), where θ (sin τ)2−n ξ(θ)= dτ, p(cos τ) Zπ/2 and suppose that ξ(0) = and ξ(π)= . Then (50) can be written in the form −∞ ∞ n 2 2 z′(ξ) 2dξ + µ 1+ z(ξ) 2 p(cos θ(ξ)) (sin θ(ξ))n−2 dξ R1 | | − 2 R1 | | λ = inf Z Z , z z(ξ) 2 p(cos θ(ξ)) q(cos θ(ξ))(sin θ(ξ))2(n−2) dξ R1 | | Z where θ(ξ) is the inverse function of ξ(θ). By Theorem 1 in [M10], 1 θ(ξ+d+δ) + p(cos θ) (sin θ)n−2 dθ δ Zθ(ξ−d−δ) λ inf ( + ) . (51) ∼ ξ∈R1 θ ξ d d>0,δ>0 q(cos θ) (sin θ)n−2 dθ Zθ(ξ−d)

12 Here the equivalence a b means that c1b a c2b, where c1 and c2 are positive constants depending only∼ on µ and n. Hence≤ (49)≤ holds with a positive Λ if and only if the infimum (51) is positive. Remark 8 In the case n = 2 the best constant in (49) is equal to 2π y′(ϕ) 2 + µ2 y(ϕ) 2 p(sin ϕ)dϕ | | | | λ := inf Z0 , (52) y 2π y(ϕ) 2q(sin ϕ)dϕ | | Z0 where the infimum is taken over all smooth functions on the interval [0, 2π]. Note that (33) is a particular case of (52) with µ =1/2 and p(t)= t. As another application of (52) we obtain the following special case of inequality (49) with n = 2. For all u C∞(R2) the inequality ∈ 0 2 2 x2 u(x) dx 1 2 2 2 | | x2 u(x) dx (53) R2 π x 2 ≤ 2 R2 |∇ | Z 2 2 1 Z (x1 + x2) arcsin 2 2 1/2 4 − (x1 + x2) holds, where 1/2 is the best constant. In order to prove (53), we choose p(t)= t2 and µ = 1 in (52), which becomes 2π y′(ϕ) 2 + y(ϕ) 2 (sin ϕ)2dϕ | | | | λ = inf Z0 , (54) y 2π y(ϕ) 2q(sin ϕ)dϕ | | Z0 where y is an arbitrary smooth 2π-periodic function. Putting η(ϕ) = y(ϕ) sin ϕ, we write (54) in the form 2π η′(ϕ) 2dϕ | | λ = inf Z0 (55) η 2π η(ϕ) 2q(sin ϕ)(sin ϕ)−2dϕ | | Z0 with the infimum taken over all 2π-periodic functions satisfying η(0) = η(π) =0. Let (sin ϕ)2 q(sin ϕ)= . ϕ(π ϕ) − In view of the well-known sharp inequality 1 z(t) 2 1 1 | | dt z′(t) 2dt t(1 t) ≤ 2 | | Z0 − Z0 (see [HLP], Th. 262), we have λ = 2 in (55). Therefore, (49) becomes (53).

4 Generalized Hardy-Sobolev inequality with sharp constant

Let Ω denote an open set in Rn and let p [1, ). By (p,a)-capacity of a compact set K Ω we mean the set function ∈ ∞ ⊂ p −a ∞ capp,a(K, Ω) = inf u x dx : u C0 (Ω), u 1 on K . Ω |∇ | | | ∈ ≥ nZ o

13 Rn In the case a =0,Ω= we write simply capp(K). The following inequality is a particular case of a more general one obtained in [M8], where Ω is an open subset of an arbitrary Riemannian manifold and Φ(x, u(x) plays the role of u(x) x −a/p. | ∇ | |∇ | | | Theorem 4 (see [M3] for q = p and [M8] for q p) ≥ (i) Let q p 1 and let Ω be an open set in Rn. Then for an arbitrary u C∞(Ω), ≥ ≥ ∈ 0 ∞ 1/q 1/p q/p cap (M , Ω) d(tq) u(x) p x −adx , (56) p,a t ≤ Ap,q |∇ | | | Z0 ZΩ where M = x Ω: u(x) t and t { ∈ | |≥ } pq 1/p−1/q Γ q−p p,q = q q−1 (57) A Γ Γ p q−p q−p for q>p, and = p(p 1)(1−p)/p. (58) Ap,p − (ii) The sharpness of this constant is checked by a sequence of radial functions in ∞ Rn C0 ( ). Moreover, there exists a radial optimizer vanishing at inﬁnity.

Being combined with the isocapacitary inequality

µ(K)γ Λ cap (K, Ω) (59) ≤ p,γ p,a where µ is a Radon measure in Ω, (56) implies the estimate

∞ 1/p γq/p 1/q µ(M ) d(tq) Λ1/p u(x) p x −adx (60) t ≤ Ap,q p,γ |∇ | | | Z0 ZΩ for all u C∞(Ω). ∈ 0 This estimate of u in the Lorentz space p/γ,q(µ) becomes the estimate in Lq(µ) for γ = p/q: L 1/p u Λ1/p u(x) p x −adx . k kLq(µ) ≤ Ap,q p,γ |∇ | | | ZΩ

In the next assertion we ﬁnd the best value of Λp,γ in (59) for the measure µ = µb deﬁned by (8).

Lemma 2 Let an 1 p < n, 0 a

14 Proof. Introducing spherical coordinates (r, ω) with r> 0 and ω Sn−1, we have ∈ ∞ 2 p ∂u 1 2 2 n−1−a capp,a(K) = inf + 2 ωu r drdsω. (63) u ≥1 Sn−1 0 ∂r r |∇ | K Z Z

1/κ Let us put here r = ρ , where n p a κ = − − n p − and y = (ρ,ω). The mapping (r, ω) (ρ,ω) will be denoted by σ. Then (63) takes the form → 2 p κp−1 ∂u κ −2 2 2 capp,a(K)= inf + ( ρ) ωu dy, (64) v Rn ∂ρ |∇ | Z −1 κ where the inﬁmum is taken over all v = u σ . Since 0 1 owing to the conditions p

p−1 p n−p n−p n p n−1 n cap (σ(K)) − S n n n mes (σ(K) . (66) p ≥ p 1 | | n − Clearly, 1 dy µ (K)= b κ y α Zσ(K) | | with n b b(n p) an α = n − = − − 0. (67) − κ n p a ≥ − − Furthermore, one can easily check that

1− α n α α n n−1 1− n µ (K) S n mes (σ(K) (68) b ≤ n b | | n − (see, for instance, [M8], Example 2.2). Combining (68) with (66), we ﬁnd

b−p−a n−p−a p−1 n−1 n−b n−b p 1 S µb(K) − | | n−p−a capp(σ(K)) (69) ≤ n p n−b − (n b) − which together with (65) completes the proof of (62). The main result of this section is as follows.

Theorem 5 Let conditions (61) hold and let q p. Then for all u C∞(Rn) ≥ ∈ 0 ∞ (n−p−a)q 1 1 q p (n−b)p q p dx µb(Mt) d(t ) p,q,a,b u(x) a , (70) ≤C Rn |∇ | x Z0 Z | | where pq Γ 1 − 1 1− 1 n p+a−b q−p p q p 1 p Γ( 2 ) (n−b)p p,q,a,b = q q−1 − n n−p−a . (71) C Γ Γ p n p a 2π 2 (n b) p+a−b q−p q−p − − − The constant (71 ) is best possible which can be shown by constructing a radial opti- ∞ Rn mizing sequence in C0 ( ).

15 Proof. Inequality (70) is obtained by substitution of (57) and (62) into (60). The sharpness of (71) follows from the part (ii) of Theorem 4 and from the fact that the isocapacitary inequality (62) becomes equality for balls. The last theorem contains the best constant in the Il’in inequality (9) as a particular case q = (n b)p/(n p a). We formulate this as the following assertion. − − − Corollary 3 Let conditions (61) hold. Then for all u C∞(Rn) ∈ 0 n−p−a n−b 1 (n−b)p dx dx p (n−p−a) p u(x) b p,a,b u(x) a , (72) Rn | | x ≤C Rn |∇ | x Z | | Z | | where p(n−b) 1− 1 n p+a−b Γ p−b p 1 p Γ( 2 ) (n−b)p p−b p(n−b) p,a,b = − n p a . n − − n−b (n−b)(p−1) C n p a 2π 2 (n b) p+a−b Γ Γ1+ − − − p−b p−b This constant is best possible which can be shown by construc ting a radial optimizing ∞ Rn sequence in C0 ( ).

5 Hardy’s inequality with sharp Sobolev remainder term

Theorem 6 For all u C∞(Rn ), u =0 on Rn−1, the sharp inequality ∈ + 2 n/(n+1) 2 2 1 u π (n 1) −1/(n+1) 2 dx + x u n u dx | | − n L n (R ) (73) n n 2 n 2/(n+1) 2( +1) + R |∇ | ≥ 4 R xn k k n−1 Z + Z + 4 Γ 2 +1 holds. Proof. We start with the Sobolev inequality

2 2 w dz w n+1 (74) n+1 L 2(n+1) (R ) Rn+1 |∇ | ≥ S k k Z n−1 with the best constant π(n+2)/(n+1)(n2 1) n+1 = − (75) S n/(n+1) n 2/(n+1) 4 Γ 2 +1 (see Rosen [R], Aubin [Au], and Talenti [Ta]). Let us introduce the cylindrical coordinates (r, ϕ, x′), where r 0, ϕ [0, 2π), and x′ Rn−1. Assuming that w does not depend on ϕ, we write (74)≥ in the∈ form ∈ ∞ 2 ∂w 2 ′ 2π + x′ w r drdx Rn−1 0 ∂r |∇ | Z Z ∞ (n−1)/(n+1) (n−1)/(n+1) 2(n+1)/(n−1) ′ (2π) n+1 w r drdx . ≥ S Rn−1 0 | | Z Z Replacing r by xn, we obtain (n−1)/(n+1) 2 −2/(n+1) 2(n+1)/(n−1) w xn dx (2π) n+1 w xn dx . Rn |∇ | ≥ S Rn | | Z + Z + 1/2 It remains to set here w = xn v and to use (75).

16 References

[Ad] Adimurthi, Hardy-Sobolev inequalities in H1(Ω) and its applications, Comm. Contem. Math. 4 (2002), no. 3, 409-434. [ACR] Adimurthi, N. Chaudhuri, M. Ramaswamy, An improved Hardy-Sobolev inequality and its applications, Proc. Amer. Math. Soc. 130 (2002), no. 489- 505. [AGS] Adimurthi, M. Grossi, S. Santra, Optimal Hardy-Rellich inequalities, maxi- mum principle and related eigenvalue problems, J. Funct. Anal. 240 (2006), no.1, 36-83. [Au] T. Aubin, Problèmes isopérimètriques et espaces de Sobolev, C.R. Acad. Sci. Paris, 280 (1975), no. 5, 279-281. [BFT1] G. Barbatis, S. Filippas, A. Tertikas, Series expansion for Lp Hardy inequalities, Indiana Univ. Math. J. 52 (2003), no. 1, 171-190. [BFT2] G. Barbatis, S. Filippas, A. Tertikas, A unified approach to improved Lp Hardy inequalities with best constants, Trans. Amer. Math. Soc. 356 (2004), no. 6, 2169-2196.

[BFL] R.D. Benguria, R.L. Frank, M. Loss, The sharp constant in the Hardy- Sobolev-Maz’ya inequality in the three dimensional upper half-space, arXiv: math/0705 3833. [BCT] B. Brandolini, F. Chiacchio, C. Trombetti, Hardy inequality and Gaussian measure, Comm. Pure Appl. Math. 6 (2007), no. 2, 411-428.

[BM] H. Brezis, M. Marcus, Hardy’s inequality revisited, Ann. Scuola Norm Pisa, 25 (1997), 217-237. [BV] H. Brezis, J.L. Vázquez, Blow-up solutions of some noinlinear elliptic problems, Rev. Mat. Univ. Comp. Madrid 10 (1997), 443-469. [CW] F. Catrina, Z.-Q. Wang, On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extremal functions, Comm. Pure Appl. Math. 54 (2001), 229-258. [CM] O. Costin, V. Maz’ya, Sharp Hardy-Leray inequality for axisymmetric divergence-free fields, arXiv:math/0703116, to appear in Calculus of Vari- ations and Partial Diff. Eq. [Da] E.B. Davies, A review of Hardy inequalities, The Maz’ya Anniversary Col- lection, Vol. 2 (Rostock, 1998), 55–67, Oper. Theory Adv. Appl., 110, Birkhäuser, Basel, 1999. [DD] J. Dávila, L. Dupaigne, Hardy-type inequalities, J. Eur. Math. Soc. 6 (2004) no. 3, 335-365. [DNY] J. Dou, P. Niu, Z. Yuan, A Hardy inequality with remainder terms in the Heisenberg group and weighted eigenvalue problem, J. Ineq. Appl. 2007 (2007), Article ID 32585, 24 pages. [E] S. Eilertsen, On weighted positivity and the Wiener regularity of a boundary point for the fractional Laplacian, Ark. Mat. 38 (2000), 53-75.

17 [EL] W. E. Evans, R.T, Lewis, Hardy and Rellich inequalities with remainders, J. Math. Ineq. 1 (2007), no. 4, 473-490. [FF] H. Federer, W.H. Fleming, Normal and integral currents, Ann. Math. 72 (1960), 458-520. [FMT1] S. Filippas, V. Maz’ya, A. Tertikas, Sharp Hardy-Sobolev inequalities, C.R. Acad. Sci. Paris, 339 (2004), no. 7, 483-486. [FMT2] S. Filippas, V. Maz’ya, A. Tertikas, On a question of Brezis and Marcus, Calc. Var. Partial Differential Equations, 25 (2006), no. 4, 491–501. [FMT3] S. Filippas, V. Maz’ya, A. Tertikas, Critical Hardy-Sobolev inequalities J. de Math. Pure et Appl., 87 (2007), 37-56. [FT] S. Filippas, A. Tertikas, Optimizing improved Hardy inequalities, J. Funct. Anal. 192 (2002), 186-233. [FTT] S. Filippas, A. Tertikas, J. Tidblom, On the structure of Hardy-Sobolev- Maz’ya inequalities, arXiv: math/0802.0986. [FS] R. L. Frank, R. Seiringer, Non-linear ground state representations and sharp Hardy inequalities, arXiv: math/0803.0503. [G] E. Gagliardo, Ulteriori proprietàdi alcune classi di funzioni in piùvariabili, Ric. Mat. 8 (1959), no. 1, 24-51. [GMGT] V. Glazer, A. Martin, H. Grosse, W. Thirring, A family of optimal conditions for the absence of bound states in a potential, in Studies in Mathe- matical Physics, E.H. Lieb, B. Simon and A.S. Wightman, eds, Princeton University Press, 1976, 169-194. [GGM] F. Gazzola, H.-Ch. Grunau, E. Mitidieri, Hardy inequalities with optimal constants and remainder terms, Trans. Amer. Math. Soc., 356 (2004), 2149- 2168. [HLP] G.H. Hardy, J.E. Littlewood, G. Polya, Inequalities, Cambridge University Press, 1952. [HHL] M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, A. Laptev, A geometrical version of Hardy’s inequality, J. Funct. Anal. 189 (2002), 539-548. [Il] V.P. Il’in, Some integral inequalities and their applications in the theory of differentiable functions of several variables, Matem. Sbornik, 54 (1961), no. 3, 331-380. [L] E.H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. Math. 118 (1983), 349-374. [MM] S. Mayboroda, V. Maz’ya, Boundedness of the Hessian of a biharmonic function in a convex domain, arXiv:math/0611058, to appear in Comm. Part. Diff. Eq. [M1] V. Maz’ya, Classes of domains and imbedding theorems for function spaces. Soviet Math. Dokl. 133 (1960), no.1, 882–885. [M2] V. Maz’ya, On a degenerating problem with directional derivative, Mat. Sb. 87 (1972), 417–454; English translation: Math. USSR Sbornik, 16 (1972), no. 3, 429-469.

18 [M3] V. Maz’ya, On certain integral inequalities for functions of many variables, Problemy Matem. Analiza, no.3 (1972), 33-68; English translation: J. of Mathematical Sciences, 1 (1973), no. 2, 205-234. [M4] V. Maz’ya, Behaviour of solutions to the Dirichlet problem for the biharmonic operator at a boundary point. Equadiff IV (Proc. Czechoslovak Conf. Differential Equations and their Applications, Prague, 1977), pp. 250–262, Lecture Notes in Math., 703, 1979, Springer, Berlin. [M5] V. Maz’ya, Sobolev Spaces, Springer, 1985. [M6] V. Maz’ya, On the Wiener-type regularity of a boundary point for the poly- harmonic operator, Appl. Anal. 71 (1999), 149–165. [M7] V. Maz’ya, The Wiener test for higher order elliptic equations, Duke Math. J., 115 (2002), no. 3, 479–512. [M8] V. Maz’ya, Lectures on isoperimetric and isocapacitary inequalities in the theory of Sobolev spaces, Contemporary Mathematics, 338 (2003), 307-340. [M9] V. Maz’ya, Conductor and capacitary inequalities for functions on topological spaces and their applications to Sobolev type imbeddings, J. Funct. Anal., 224 (2005), no. 2, 408–430. [M10] V. Maz’ya, Conductor inequalities and criteria for Sobolev type two-weight imbeddings, J. Comp. Appl. Math., 194 (2006), no. 1, 94–114. [MT] V. Maz’ya, G. Tashchiyan, On the behavior of the gradient of the solution of the Dirichlet problem for the biharmonic equation near a boundary point of a three-dimensional domain. (Russian) Sibirsk. Mat. Zh. 31:6, 113–126; English translation: Siberian Math. J. 31:6, 970–983 (1991). [MP] S. Mikhlin, S. Prössdorf, Singular Integral Operators, Springer, 1986. [Na] A.I. Nazarov, Hardy-Sobolev inequalities in a cone, J. Math. Sci. 132 (2006), no. 4, 419-427. [N] L. Nirenberg, On elliptic partial differential equations, Lecture II, Ann. Sc. Norm. Sup. Pisa, Ser. 3 13 (1959), 115-162.

[R] G. Rosen, Minimum value for C in the Sobolev inequality ϕ3 C gradϕ 3, SIAM J. Appl. Math. 21:1 (1971), 30-33. k k ≤ k k [SW] E.M. Stein, G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, 1971. [Ta] G. Talenti, Best constant in Sobolev inequality. Ann. Mat. Pura Appl., 136 (1976), 353–172. [TT] A. Tertikas, K. Tintarev, On existence of minimizers for the Hardy-Sobolev- Maz’ya inequality, Ann. Mat. Pura Appl. 186 (2007), no. 4, 645-662. [TZ] A. Tertikas, N.B. Zographopoulos, Best constants in the Hardy-Rellich inequalities and related improvements, Adv. Math. 209 (2007), no. 2, 407-459.

1,p [Ti1] J. Tidblom, A geometrical version of Hardy’s inequality for W0 , Proc. Amer. Math. Soc. 138 (2004), no. 8, 2265-2271.

19 [Ti2] J. Tidblom, A Hardy inequality in the half-space, J. Funct. Anal. 221 (2005), 482-495. [VZ] J.L. V´azquez, E. Zuazua, The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential, J. Funct. Anal. 173 (2000), 103-153. [YZ] S. Yaotian, C. Zhihui, General Hardy inequalities with optimal constants and remainder terms, J. Ineq. Appl. no. 3 (2005), 207-219.