TY - JOUR

T1 - When does the norm of a Fourier multiplier dominate its L∞ norm?

AU - Karlovich, Alexei

AU - Shargorodsky, Eugene

N1 - info:eu-repo/grantAgreement/FCT/5876/147204/PT#
sem pdf conforme despacho

PY - 2018

Y1 - 2018

N2 - One can define Fourier multipliers on a Banach function space using the direct and inverse Fourier transforms on L2(ℝn) or using the direct Fourier transform on S(ℝn) and the inverse one on S(ℝn). In the former case, one assumes that the Fourier multipliers belong to L∞(ℝn), while in the latter one this requirement may or may not be included in the definition. We provide sufficient conditions for those definitions to coincide as well as examples when they differ. In particular, we prove that if a Banach function space X(ℝn) satisfies a certain weak doubling property, then the space of all Fourier multipliers Mx(ℝn) is continuously embedded into L∞(ℝn) with the best possible embedding constant one. For weighted Lebesgue spaces Lp(ℝn, w), the weak doubling property is much weaker than the requirement that w is a Muckenhoupt weight, and our result implies that (Formula presented.) for such weights. This inequality extends the inequality for n=1 from Theorem of [Berkson and Gillespie, Bull. Sci. Math. 122 (1998) 427–454], where it is attributed to J. Bourgain. We show that although the weak doubling property is not necessary, it is quite sharp. It allows the weight w in Lp(ℝn, w) to grow at any subexponential rate. On the other hand, the space L,p(ℝex has plenty of unbounded Fourier multipliers.

AB - One can define Fourier multipliers on a Banach function space using the direct and inverse Fourier transforms on L2(ℝn) or using the direct Fourier transform on S(ℝn) and the inverse one on S(ℝn). In the former case, one assumes that the Fourier multipliers belong to L∞(ℝn), while in the latter one this requirement may or may not be included in the definition. We provide sufficient conditions for those definitions to coincide as well as examples when they differ. In particular, we prove that if a Banach function space X(ℝn) satisfies a certain weak doubling property, then the space of all Fourier multipliers Mx(ℝn) is continuously embedded into L∞(ℝn) with the best possible embedding constant one. For weighted Lebesgue spaces Lp(ℝn, w), the weak doubling property is much weaker than the requirement that w is a Muckenhoupt weight, and our result implies that (Formula presented.) for such weights. This inequality extends the inequality for n=1 from Theorem of [Berkson and Gillespie, Bull. Sci. Math. 122 (1998) 427–454], where it is attributed to J. Bourgain. We show that although the weak doubling property is not necessary, it is quite sharp. It allows the weight w in Lp(ℝn, w) to grow at any subexponential rate. On the other hand, the space L,p(ℝex has plenty of unbounded Fourier multipliers.

UR - http://www.scopus.com/inward/record.url?scp=85054680604&partnerID=8YFLogxK

U2 - 10.1112/plms.12206

DO - 10.1112/plms.12206

M3 - Article

AN - SCOPUS:85054680604

JO - Proceedings of the London Mathematical Society

JF - Proceedings of the London Mathematical Society

SN - 0024-6115

ER -