By Luca Capogna

Contemporary advancements in geometric degree thought and harmonic research have ended in new and deep effects about the regularity of the help of measures which behave "asymptotically" (for balls of small radius) because the Euclidean quantity. A remarkable function of those effects is they really signify flatness of the help when it comes to the asymptotic habit of the degree. Such characterizations have resulted in vital new growth within the learn of harmonic degree for non-smooth domain names. This quantity offers an up to date review and an advent to the study literature during this zone. The presentation follows a chain of 5 lectures given by means of Carlos Kenig on the 2000 Arkansas Spring Lecture sequence on the college of Arkansas. the unique lectures were multiplied and up-to-date to mirror the speedy development during this box. A bankruptcy at the planar case has been further to supply a old point of view. extra historical past has been incorporated to make the fabric available to complex graduate scholars and researchers in harmonic research and geometric degree conception

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28. Let z be a nonzero complex number with nonnegative real part. Then, 2 π d2 − |ξ|2 e 4z F e−z|·| (ξ) = z def with z − 2 = |z|− 2 e−i 2 θ if z = |z|eiθ with θ ∈ [−π/2, π/2]. d d d Proof. Let us remark that for any ξ in Rd , the functions z −→ e−i(x|ξ) e−z|x| dx 2 z −→ and Rd π z d 2 e− |ξ|2 4z are holomorphic on the domain D of complex numbers with positive real part. 20) states that these two functions coincide on the intersection of the real line with D. Thus, they also coincide on the whole domain D.

67. There exist two constants, c and C, depending only on the d dimension d, such that for any function u ∈ H 2 (Rd ), we have exp c Rd |f (x)| f d2 2 − 1 dx ≤ C. H Proof. As usual, arguing by density and homogeneity, it suﬃces to consider the case where f is in S and satisﬁes f d2 = 1. 30) and the deﬁnition of nonhomogeneous Sobolev spaces, there exists some constant Cd (depending only on the dimension d) such that √ f L2p ≤ Cd p for all p ≥ 1. 41) For all x ∈ Rd , we may write exp c|f (x)|2 − 1 = p≥1 cp |f (x)|2p .

33) The estimate of T f in terms of the B˙ −σ norm is described by the following proposition. 46. For σ ∈ ]0, d], a constant C exists such that Tf B˙ −σ ≤ 2d−2σ f B˙ −σ +C f L1 . Proof. 32), we have λd−σ θ(λ·) (T f ) L∞ ≤ λd−σ θ Tf L∞ L1 ≤ λd−σ θ L∞ f L1 , we get sup λ−σ λd θ(λ·) (T f ) L∞ ≤ θ L∞ f L1 . 34) λ≤1 The case where λ is large (which corresponds to high frequencies) is more intricate. We ﬁrst estimate λd (θ(λ·) (T f ))(x) when x is not too close to T (Q), def namely, x ∈ Qc = {x ∈ Q / d(x, T (Q)) ≥ 1/8}.