HARMONIC ANALYSIS 2016

EXERCISES VI (POISSON KERNEL, )

MATANIA BEN-ARTZI

1. BOOKS [Co] R. Courant, Differential and Integral Calculus, Vol. I ,Blackie and Son, 1934.

[Ka] Y. Katznelson, An Introduction to Harmonic Analysis, Wiley, 1968.

[Ru] W. Rudin,Functional Analysis, McGraw-Hill Co. 1966.

[W] H.F.Weinberger, A First Course in Partial Differential Equations, Wiley Publ. 1965.

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Notation B(y, R) The open ball (in Rn) of radius R, centered at y. S(Rn) is the Schwartz space of rapidly decaying smooth functions. The seminorms on S are defined by α β n pα,β(f) = max |x D f(x)|, α, β ∈ . n N x∈R S0(Rn) is the space of continuous linear functionals on S(Rn). Such functionals are called tempered distributions. p n 0 n R If f ∈ L ( ), 1 ≤ p ≤ ∞, then Tf ∈ S ( ) is defined by Tf (φ) = n f(x)φ(x)dx, φ ∈ R R R S(Rn). Z − n −iξx fb(ξ) = Ff(ξ) = (2π) 2 f(x)e dx. n R ***************************************************************************************** (1) Let φ ∈ S(Rn) and T ∈ S0(Rn). Prove that for every j ∈ {1, 2, ..., n} ,

Dj(φT ) = (Djφ)T + φDjT. (2) In this problem n = 1. Let f(x) = ex, g(x) = ex cos(ex), x ∈ R. 0 Prove that Tf ∈/ S (R) but Tg ∈ S(R). (See [Ru], Exercises Ch.7, p. 187).

Date: March 24, 2017. 1 2 MATANIA BEN-ARTZI

(3) In this problem n = 1. Prove that the functional Z φ(x) T φ = PV dx, φ ∈ S( n), x R R d is a tempered distribution and compute the F[ dx T ]. (4) (Poisson kernel ) Definition. The kernel 1 1 P (x) = , x ∈ , π x2 + 1 R −1 x is called the Poisson kernel. For every ε > 0 we set Pε(x) = ε P ( ε ) = 1 ε π x2+ε2 . (a) Show that the family {Pε}ε>0 is a positive summability kernel. (b) Conclude that (for ϕ ∈ S), 1 Z ε ϕ(τ) = lim ϕ(ξ)dξ, τ ∈ R, π ε→0+ (ξ − τ)2 + ε2 R and the convergence is uniform in τ ∈ R. (c) Extend this result to any bounded continuous function g(ξ). Give a condition on g that will ensure that the limit is attained uniformly in τ ∈ R. (5) ( harmonic functions in the upper half-plane) Given a bounded continuous function g(ξ), ξ ∈ R, define 1 Z y u(x, y) = g(ξ)dξ, (x, y) ∈ × (0, ∞). π (ξ − x)2 + y2 R R (a) Prove that u(x, y) is harmonic in the upper half plane. (b) Prove that u(x, y) is bounded in the upper half-plane and determine its upper and lower bounds in terms of sup g and inf g. R R (c) Assume now that g ∈ L1(R) ∩ L∞(R). Using the complex notation z = x + iy prove that the function Z g(ξ) G(z) = dξ ξ − z R is analytic in the upper half-plane y > 0. (d) Is there a connection between G(z) and u(x, y)? (6) Reminder: For ψ ∈ S the transformation 1 Z ψ(ξ) ψ(τ) = PV dξ, τ ∈ , H πi ξ − τ R R is the Hilbert transform of ψ. Define the kernel 1 t Peε(t) = , ε > 0 π t2 + ε2 (which is sometimes called the ”conjugate Poisson kernel”). ˜ 2 (a) Show that if ϕ ∈ S then Pε ? ϕ ∈ L (R) and −ε|ξ| −ε|ξ| F[Peε ? ϕ](ξ) = −i(H − RH)(ξ)e ϕˆ(ξ) = −i sgn(ξ)e ϕˆ(ξ), ξ ∈ R, where H(x) is the Heaviside function and Rθ(x) = θ(−x) is the reflec- tion operation. HARMONIC ANALYSIS 2016-EXERCISES VI 3

2 Conclude that the convolution can be extended so that Peε ? g ∈ L (R) for any g ∈ L2(R). (b) Prove that 2 Peε ? g −−−−→ −iHg, for any g ∈ L (R). ε→0+ (c) Prove that 2 Peε ? g = −iPε ? Hg, for any g ∈ L (R). (Suggestion: Look at Fourier transforms).

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