Sharp Weak Type Inequalities for Fractional Integral Operators

Suppose that d≥1 is an integer, α∈(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 . Let p, q be arbitrary numbers satisfying the conditions 1≤p<d/α and 1/q=1/p?α/d. We determine the optimal constant K, which depends on α, d and p, such that for any f∈L ^p ((0,1] ^d ) we have

$$ ||I_{\alpha } f||_{L^{q,\infty }((0,1]^{d})}\leq K||f||_{L^{p}((0,1]^{d})}. $$

In fact, we shall prove this inequality in the more general context of probability spaces equipped with a regular tree-like structures. This allows us to obtain this result also for non-integer dimension. The proof exploits a certain modification of the so-called Bellman function method and appropriate interpolation-type arguments. We also present a sharp weighted weak-type bound for I _α , which can be regarded as a version of the Muckenhoupt-Wheeden conjecture for fractional integral operators.