In this paper, we introduce the sharp maximal function in this general setting, and establish the equivalence of theLpnorms between the sharp maximal function and the Hardy Littlewood maximal function, as well the John Nirenberg type inequalities.
In particular, we give new proofs, which completely avoid the good-λ inequalities, of Coifman's inequality relating singular integrals and the maximal operator, of the Fefferman Stein inequality relating the maximal operator and the sharp maximal operator, and the Muckenhoupt–Wheeden inequality relating the fractional integral operator and the fractional maximal operator.
Similarly, the sharp maximal operator M s ♯ is defined by.
They first established the sharp maximal estimates, then the end-point estimates were acquired.
The sharp maximal operator is an analogue of the Hardy-Littlewood maximal operator, which satisfies.
The (Hardy-Littlewood) maximal and the sharp maximal functions M f, f p, resp.
For and, the sharp maximal function on is defined by (2.1).
The main purpose of this paper is to prove the sharp maximal inequalities for the commutator.
In Section 3, the sharp maximal inequalities with Orlicz norms applied to k-quasiminimizer are obtained.
Similarly, for a locally -integrable form, we define the sharp maximal operator by (1.18).
In Section 2, we introduce the sharp maximal operator (M^), associated with (K_{B,S}) and prove Lemma 2.6.
