Maximal inequalities related to generalized a.e. continuity

Abstract

An integral inequality of the classical Hardy-Littlewood type is obtained for the maximal function of positive convolution operators associated with approximations of the identity in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R Superscript n"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>R</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{R^n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . It is shown that the (formally) rearranged maximal function can in general be estimated by an elementary integral involving the decreasing rearrangements of the kernel of the approximation and the function being approximated. (The estimate always holds when the kernel has compact support or a decreasing radial majorant integrable in a neighborhood of infinity; a one-dimensional counterexample shows that integrability alone may not suffice.) The finiteness of the integral determines a Lorentz space of functions which are a.e. continuous in the generalized sense of the approximation. Conversely, in dimension one it is established that this space is the largest strongly rearrangement invariant Banach space of such functions. In particular, the new inequality provides access to the study of Cesàro continuity of order less than one.