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“Sublinear” Sobolev Inequality of the Form ( A SUBLINEAR SOBOLEV INEQUALITY FOR p-SUPERHARMONIC FUNCTIONS NGUYEN CONG PHUC Abstract. We establish a \sublinear" Sobolev inequality of the form n−q 1 nq q nq q u n−q dx ≤ C jDuj dx ˆ n ˆ n R R n for all global p-superharmonic (1 < p < 2) functions u in R , n ≥ 2, n with infR u = 0 and p − 1 < q < 1. The same result also holds for the class of A-superharmonic functions. More general sublinear trace inequalities, where Lebesgue measure is replaced by a general measure, are also considered. 1. Introduction A celebrated inequality due to S. L. Sobolev [Sob] states that for any 1 < q < n there is a constant C = C(n; q) > 0 such that for every function 1 n u 2 C0 (R ) it holds that n−q 1 nq nq q (1.1) u n−q dx ≤ C jDujqdx : ˆRn ˆRn 1 n Inequality (1.1) also holds for all u 2 C0 (R ) in the case q = 1 by the Gagliardo{Nirenberg estimate (see [Gag, Nir]). However, it seems to be an 1 n open problem whether (1.1) holds for all u 2 C0 (R ) and for all or just certain values of q in the \sublinear" range q 2 (0; 1). The main purpose of this note is to show that Sobolev inequality of the form (1.1) remains to hold for certain exponent q < 1 when restricted to n a class of positive p-superharmonic (1 < p < 2) functions u in R , n ≥ 2, with a mild decay at infinity. Precisely, it will be shown that for 1 < p < 2, inequality (1.1) holds with q 2 (p − 1; 1) for all p-superharmonic functions n u in R , n ≥ 2, such that infRn u = 0 (see Theorem 1.4 below). More generally, we also consider sublinear trace inequalities of Sobolev type for p-superharmonic functions, where Lebesgue measure on the left-hand side is replaced by a general measure (see Theorem 3.1). We remark that when 1 n 1 < p ≤ 2 − 1=n the `gradient' Du in (1.1) may not belong to Lloc(R ) and thus should be understood by means of an approximation (see (1.4) below). The same kind of Sobolev inequality also holds for the more general class of positive A-superharmonic functions. We now recall the notion of p-superharmonicity and A-superharmonicity. Let 1 < p < n, n ≥ 2, and let n n n A : R × R ! R be a vector valued mapping which satisfies the following 1 2 NGUYEN CONG PHUC conditions: n the mapping x !A(x; ξ) is measurable for all ξ 2 R ; n the mapping ξ !A(x; ξ) is continuous for a:e: x 2 R ; n and there are constants 0 < α ≤ β < 1 such that for a.e. x in R , and for n all ξ in R , (1.2) A(x; ξ) · ξ ≥ α jξjp ; jA(x; ξ)j ≤ β jξjp−1 ; [A(x; ξ1) − A(x; ξ2)] · (ξ1 − ξ2) > 0; if ξ1 6= ξ2 : 1; p For u 2 Wloc (Ω), where Ω is an open set, we define the divergence of 1 A(x; ru) in the sense of distributions, i.e., if ' 2 C0 (Ω), then divA(x; ru)(') = − A(x; ru) · r' dx: ˆΩ 1; p It is well known that every solution u 2 Wloc (Ω) to the equation −divA(x; ru) = 0(1.3) has a continuous representative. Such continuous solutions are said to be 1; p A-harmonic in Ω. If u 2 Wloc (Ω) and A(x; ru) · r' dx ≥ 0; ˆΩ 1 for all nonnegative ' 2 C0 (Ω), i.e., −divA(x; ru) ≥ 0 in the distributional sense, then u is called a supersolution to (1.3) in Ω. A lower semicontinuous function u :Ω ! (−∞; 1] is called A-super- harmonic if u is not identically infinite in each component of Ω, and if for all open sets D such that D ⊂ Ω, and all functions h 2 C(D), A-harmonic in D, it follows that h ≤ u on @D implies h ≤ u in D. In the special case A(x; ξ) = jξjp−2ξ, A-superharmonicity is often referred to as p-superharmonicity. It is worth mentioning that the latter can also be defined equivalently using the language of viscosity solutions (see [JLM]). We recall here the fundamental connection between supersolutions of (1.3) and A-superharmonic functions [HKM]. Proposition 1.1 ([HKM]). (i) If v is A-superharmonic on Ω then v(x) = ess lim inf v(y); x 2 Ω: y!x 1; p Moreover, if v 2 Wloc (Ω) then −divA(x; rv) ≥ 0: 1; p (ii) If u 2 Wloc (Ω) is such that −divA(x; ru) ≥ 0; then there is an A-superharmonic function v such that u = v a.e. 1; p (iii) If v is A-superharmonic and locally bounded, then v 2 Wloc (Ω) and −divA(x; rv) ≥ 0: A SUBLINEAR SOBOLEV INEQUALITY 3 Note that an A-superharmonic function u does not necessarily belong 1; p to Wloc (Ω), but its truncation minfu; kg does for every integer k due to Proposition 1.1(iii). Using this we let Du stand for the a.e. defined function (1.4) Du = lim r ( minfu; kg): k!1 1 1; 1 If either u 2 L (Ω) or u 2 Wloc (Ω), then Du coincides with the regular distributional gradient ru of u. In general, we have the following gradient estimates [KM1] (see also [HKM]). Proposition 1.2 ([KM1]). Suppose u is A-superharmonic in Ω, 0 < r < n(p−1) n r s n−1 , and 0 < s < n−1 . Then Du 2 Lloc(Ω) and A(x; Du) 2 Lloc(Ω). 1 Moreover, if p > 2 − n , then Du coincides with the distributional gradient ru of u. We can now extend the definition of the divergence of A(x; ru) to those u which are merely A-superharmonic in Ω. For such u we set −divA(x; ru)(') = −divA(x; Du)(') = A(x; Du) · r' dx ˆΩ 1 for all ' 2 C0 (Ω). Note that by Proposition 1.2 and the dominated con- vergence theorem, −divA(x; ru)(') = lim A(x; r minfu; kg) · r' dx ≥ 0 k!1 ˆΩ 1 whenever ' 2 C0 (Ω) and ' ≥ 0. Since −divA(x; ru) is a nonnegative distribution in Ω for an A-super- harmonic u, it follows that there is a nonnegative (not necessarily finite) Radon measure denoted by µ[u] such that −divA(x; ru) = µ[u] in Ω: An important contribution to the theory of A-superharmonic functions is the following pointwise estimates by Wolff's potentials obtained by T. Kilpel¨ainenand J. Mal´y. n Theorem 1.3 ([KM1, KM2]). Let u be an A-superharmonic function in R n with infRn u = 0. If −div A(x; ru) = µ, then for all x 2 R (1.5) C1W1;pµ(x) ≤ u(x) ≤ C2W1;pµ(x); where the constants C1;C2 depend only on n; p and the structural constants α; β. Here W1;pµ is the Wolff's potential of µ defined by 1 1 p−1 µ(Bt(x)) dt n W1;pµ(x) = n−p ; x 2 R : ˆ0 t t We are now ready to state the main result of the paper. 4 NGUYEN CONG PHUC Theorem 1.4. Let 1 < p < 2 and p − 1 < q < 1. For any A-superharmonic n function u in R such that infRn u = 0 we have n−q 1 nq nq q u n−q dx ≤ C jDujqdx ˆRn ˆRn with a constant C = C(n; p; q; α; β). The proof if Theorem 1.4 is presented in the next section. It relies on the upper bound in (1.5), which serves as a `representation' for A-superharmonic functions. We mention here that Wolff potential estimates for k-convex functions have recently been used as an important tool in [Ver] to give a new and purely elliptic proof of the so-called Hessian Sobolev inequality of N. S. Trudinger and X.-J. Wang. 2. Proof of Theorem 1.4 Proof. Let µ = µ[u] be the measure associated to the A-superharmonic function u. We first observe that in the case 1 < p < 2 we have 1 1 1 1 µ(Bt(x)) p−1 dt µ(Bt(x)) dt p−1 W1;pµ(x) = n−p ≤ C n−p ; ˆ0 t t ˆ0 t t and thus by Fubini's theorem we find dµ(y) 1 p−1 (2.1) W1;pµ(x) ≤ C n−p = C fIpµ(x)g : ˆRn jx − yj Here for 0 < α < n, Iαµ, stands for the Riesz's potential of order α of the measure µ defined by 1 dµ(y) n Iαµ(x) = n−α ; x 2 R ; γn(α) ˆRn jx − yj where 2απn=2Γ(α=2) γ (α) = : n Γ((n − α)=2) Using the convolution identity (see [Stei, p. 118]) γ (p − 1)γ (1) j · j−n+(p−1) ∗ j · j−n+1(z) = n n jzj−n+p; jzj= 6 0; γn(p) and Fubini's theorem (using that µ ≥ 0) we can write Ipµ = Ip−1(I1µ); and thus in view of (2.1) we get dµ(y) 1 p−1 W1;pµ(x) ≤ C n−p = C fIp−1(I1µ)(x)g : ˆRn jx − yj A SUBLINEAR SOBOLEV INEQUALITY 5 Now by Theorem 1.3 and the last inequality we have nq nq (2.2) u n−q dx ≤ C (W1;pµ(x)) n−q dx ˆRn ˆRn nq ≤ C (Ip−1(I1µ)(x)) (n−q)(p−1) dx ˆRn n q n−q ≤ C (I1µ(x)) p−1 dx ; ˆRn where we used the standard Sobolev embedding in the last inequality.
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