Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Boundedness of multilinear Littlewood--Paley operators with convolution type kernels on products of BMO spaces

Runzhe Zhang, Hua Wang

TL;DR

The paper addresses the problem of existence and boundedness for multilinear Littlewood--Paley operators on products of $\mathrm{BMO}$ spaces, showing that finiteness at a single point implies finiteness almost everywhere and that the square of these operators maps $([\mathrm{BMO}(\mathbb{R}^n)]^m)$ into $\mathrm{BLO}(\mathbb{R}^n)$ with bounds controlled by the $\mathrm{BMO}$ norms of the inputs. Using convolution-type kernels and a careful decomposition of the square functions, the authors extend linear and bilinear results to the multilinear setting, obtaining strong $L^p$ and weak-type bounds and BLO-continuity for $g(\vec{f})$, $S(\vec{f})$, and $g^{*}_{\lambda}(\vec{f})$, including the case when inputs are essentially bounded. They also show that these results persist for non-convolution type kernels, drawing on prior multilinear results and adapting the proofs to the multilinear context. Overall, the work extends the Littlewood--Paley theory in a robust way to multilinear operators on BMO spaces, producing sharp BLO-type bounds and expanding applicability to non-convolution kernels with potential PDE and analytic implications.

Abstract

In this paper, the authors establish the existence and boundedness of multilinear Littlewood--Paley operators on products of BMO spaces, including the multilinear $g$-function, multilinear Lusin's area integral and multilinear $g^{\ast}_λ$-function. The authors prove that if the above multilinear operators are finite for a single point, then they are finite almost everywhere. Moreover, it is shown that these multilinear operators are bounded from $\mathrm{BMO}(\mathbb R^n)\times\cdots\times \mathrm{BMO}(\mathbb R^n)$ into $\mathrm{BLO}(\mathbb R^n)$ (the space of functions with bounded lower oscillation), which is a proper subspace of $\mathrm{BMO}(\mathbb R^n)$ (the space of functions with bounded mean oscillation). The corresponding estimates for multilinear Littlewood--Paley operators with non-convolution type kernels are also discussed.

Boundedness of multilinear Littlewood--Paley operators with convolution type kernels on products of BMO spaces

TL;DR

The paper addresses the problem of existence and boundedness for multilinear Littlewood--Paley operators on products of spaces, showing that finiteness at a single point implies finiteness almost everywhere and that the square of these operators maps into with bounds controlled by the norms of the inputs. Using convolution-type kernels and a careful decomposition of the square functions, the authors extend linear and bilinear results to the multilinear setting, obtaining strong and weak-type bounds and BLO-continuity for , , and , including the case when inputs are essentially bounded. They also show that these results persist for non-convolution type kernels, drawing on prior multilinear results and adapting the proofs to the multilinear context. Overall, the work extends the Littlewood--Paley theory in a robust way to multilinear operators on BMO spaces, producing sharp BLO-type bounds and expanding applicability to non-convolution kernels with potential PDE and analytic implications.

Abstract

In this paper, the authors establish the existence and boundedness of multilinear Littlewood--Paley operators on products of BMO spaces, including the multilinear -function, multilinear Lusin's area integral and multilinear -function. The authors prove that if the above multilinear operators are finite for a single point, then they are finite almost everywhere. Moreover, it is shown that these multilinear operators are bounded from into (the space of functions with bounded lower oscillation), which is a proper subspace of (the space of functions with bounded mean oscillation). The corresponding estimates for multilinear Littlewood--Paley operators with non-convolution type kernels are also discussed.
Paper Structure (9 sections, 23 theorems, 159 equations)

This paper contains 9 sections, 23 theorems, 159 equations.

Table of Contents

  1. Introduction and preliminaries
  2. Linear Littlewood--Paley operators
  3. Multilinear Littlewood--Paley operators
  4. Definitions and notations
  5. Lebesgue spaces, $\mathrm{BMO}$ and $\mathrm{BLO}$ spaces
  6. Inclusion relations between $L^{\infty}$, $\mathrm{BLO}$ and $\mathrm{BMO}$
  7. Main results
  8. Proofs of Theorems \ref{['mainthm1']} and \ref{['mainthm2']}
  9. Concluding remarks

Key Result

Theorem 1.1

Suppose that $\psi\in L^1(\mathbb R^n)$ satisfies psi1, psi2 with $\delta=1$ and psi4. If $f\in \mathrm{BMO}(\mathbb R^n)$, then $g_{\psi}(f)$ is either infinite everywhere or finite almost everywhere, and in the latter case, there exists a positive constant $C>0$, independent of $f$, such that

Theorems & Definitions (32)

  • Theorem 1.1
  • Definition 1.2: shixuepeng
  • Definition 1.3: shixuepeng
  • Theorem 1.4: he
  • Theorem 1.5: he
  • Remark 2.1
  • Theorem 3.1: xueqingxuepeng
  • Theorem 3.2: shixueqing
  • Remark 3.3
  • Theorem 3.4
  • ...and 22 more