By Elias M. Stein

This e-book comprises an exposition of a few of the most advancements of the final 20 years within the following parts of harmonic research: singular fundamental and pseudo-differential operators, the speculation of Hardy areas, L\sup\ estimates regarding oscillatory integrals and Fourier critical operators, relatives of curvature to maximal inequalities, and connections with research at the Heisenberg group.

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**Sample text**

Let ???? be a real normed space and ???? be some sublinear functional on ????. Let ???? ⊂ ???? be a linear subspace, and suppose that a functional ℓ ∈ ????∗ satisfies ⟨????, ℓ⟩ ≤ ????(????) for all ???? ∈ ????. e. ⟨????, ℓ⟩̂ = ⟨????, ℓ⟩ (???? ∈ ????) and ⟨????, ℓ⟩̂ ≤ ????(????) for all ???? ∈ ????. 21 since it is beyond the scope of this book. Instead, we prove three interesting consequences of this theorem which we will need in Chapter 4. 22. Let ???? be a real normed space and ???? ⊂ ???? be a linear subspace. Then the following is true.

Max |???? (????)| . 65) are equivalent on the space ????1 ([????, ????]); this is an immediate consequence of the mean value theorem for differentiable functions. We come now to another important function class which is situated “between” 1 ???? ([????, ????]) and ????([????, ????]). 39. A function ???? : [????, ????] → ℝ is called Lipschitz continuous if there exists a constant ???? > 0 such that |????(????) − ????(????)| ≤ ????|???? − ????| (???? ≤ ????, ???? ≤ ????) . 66) More generally, ???? is called Hölder continuous (or ????-Lipschitz continuous for 0 < ???? ≤ 1) if there exists a constant ???? > 0 such that |????(????) − ????(????)| ≤ ????|???? − ????|???? (???? ≤ ????, ???? ≤ ????) .

0 L’Hospital’s rule shows that ???? is continuous at zero, and thus on the whole inter val [0, 1]. On the other hand, for any ???? > 0, we have lim ????→0+ |????(????) − ????(0)| 1 = lim ????→0+ ???????? log ???????? and so ???? ∈ ̸ Lip???? ([0, 1]) for any ???? ∈ (0, 1]. 33 for Hölder continuous (in particular, Lipschitz continuous) functions; here, the proof is even constructive. This result is usually referred to as McShane extension theorem [210]. 42 (McShane). Let ???? ⊂ ℝ and ???? : ???? → ℝ be Hölder continuous on ???? with Hölder exponent ???? ∈ (0, 1].