Abstract
Suppose that d ≥ 1 is an integer, \({\alpha \in (0,d)}\) is a fixed parameter and let Iα be the fractional integral operator associated with d-dimensional Walsh–Fourier series on [0, 1)d. The paper contains the proof of the sharp weak-type estimate $$||I_\alpha(f)||_{L^{d/(d-\alpha),\infty}([0,1)^d)}\leq\frac{2^d-1}{(2^{d-\alpha}-1)(2^\alpha-1)}||f||_{L^1([0,1)^d)}.$$ The proof rests on Bellman-function-type method: the above estimate is deduced from the existence of a certain family of special functions.
Highlights
Our motivation comes from the very natural question about sharp versions of estimates for d-dimensional Walsh system
As evidenced in numerous papers, such inequalities play an important role in many areas of mathematics, including approximation theory, Fourier analysis, harmonic analysis and probability theory
We refer the interested reader to the works [2,5,20,21,23,24] and references therein
Summary
Our motivation comes from the very natural question about sharp versions of estimates for d-dimensional Walsh system. As evidenced in numerous papers, such inequalities play an important role in many areas of mathematics, including approximation theory, Fourier analysis, harmonic analysis and probability theory. We refer the interested reader to the works [2,5,20,21,23,24] and references therein. Let us start with introducing the necessary background and notation. We will work with functions defined on the unit cube [0, 1)d in Rd, equipped with its dyadic sub-cubes, i.e., the sets of the form. Recall that the Rademacher system {rn}n≥0 of functions on [0, 1) is given by rn(t) = sgn sin(2n+1πt)
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