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Abstract

We study some operators originating from classical Littlewood–Paley theory. We consider their modification with respect to our discontinuous setup, where the underlying process is the product of a one-dimensional Brownian motion and a $d$-dimensional symmetric stable process. Two operators in focus are the $G^{*}$ and area functionals. Using the results obtained in our previous paper, we show that these operators are bounded on $L^p$. Moreover, we generalize a classical multiplier theorem by weakening its conditions on the tail of the kernel of singular integrals.