Sharp Weak Type Inequality for Fractional Integral Operators Associated with d-Dimensional Walsh–Fourier Series
Sharp Weak Type Inequality for Fractional Integral Operators Associated with d-Dimensional Walsh–Fourier Series
Suppose that d ≥ 1 is an integer, $${\alpha \in (0,d)}$$ is a fixed parameter and let I α 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 …