Boundedness of Oscillation and Variation of Semigroups in Musielak–Orlicz–Hardy Spaces

Abstract

In this paper, let $$({\cal X},d,\mu)$$ be a space of homogeneous type, L be a non-negative self-adjoint operator on $${L^2}({\cal X})$$ , and the kernels of the semigroup {(tL)me−tL}t>0 satisfy the Gaussian upper bound estimates for m ∈ ℤ+ ≔ {0} ∪ ℤ. The author shows that the oscillation operator and the variation operator of the semigroup {(tL)me−tL}t>0 are bounded from Musielak–Orlicz–Hardy space $${H_{\varphi,L}}({\cal X})$$ to Musielak–Orlicz space $${L^\varphi}({\cal X})$$ , where φ is a growth function.