Quantitative weighted estimates for some singular integrals related to critical functions (Q1979233)

This paper is concerned with quantitative weighted estimates for certain singular integrals corresponding to the new class of weights on the space of homogeneous type. The assumptions on the kernels of these singular integrals do not have any regularity conditions. Recall that \((X, d, \mu)\) is a metric space with the distance \(d\) and a nonnegative Borel doubling measure \(\mu\). The measure \(\mu\) satisfies the doubling property and the reverse doubling property. Quantitative weighted estimates for singular integrals has attracted a lot of attention. \textit{T. P. Hytönen} [Ann. Math. (2) 175, No. 3, 1473--1506 (2012; Zbl 1250.42036)] proved that, if \(T\) is a Calderón-Zygmund operator, Muckenhoupt weights \(w \in A_p\), then \[ \|T\|_{L_{w}^{p} \rightarrow L_{w}^{p}} \leq C_{T, n, p}[w]_{A_{p}}^{\max \left\{1, \frac{1}{p-1}\right\}},\ 1<p<\infty \] and the weighted estimate is sharp. By making use of a supremum of sparse operators, \textit{A. K. Lerner} [Int. Math. Res. Not. 2013, No. 14, 3159--3170 (2013; Zbl 1318.42018)] gave a simpler proof of the above estimate. In order to extend it to some weight class which is larger than the class of Muckenhoupt weights, \textit{B. Bongioanni} et al. [J. Math. Anal. Appl. 373, No. 2, 563--579 (2011; Zbl 1203.42029)] introduced a new class of weights related to the Schrödinger operator \(L\) which contains all the Muckenhoupt weights. This new class of weights is suitable to the study of the boundedness of singular integrals such as the maximal operator, the Riesz transform and the square functions associated to the Schrödinger operator \(L\). The regularity of the heat kernel of \(L\) plays an important role. \textit{F. K. Ly} [J. Math. Soc. Japan 68, No. 2, 489--533 (2016; Zbl 1348.35060)] gave the boundedness of the spectral multiplier and the generalized Riesz transform on weighted \(L^p_w\) spaces with \( w \in {A_{p}^{\rho, \theta}}\). This paper proves the quantitative weighted estimates with respect to new weights in terms of \([w]_{A_{p}^{\rho, \theta}}\) for the spectral multiplier and the generalized Riesz transform. Definition 1.1. A function \(\rho: X\rightarrow (0, \infty)\) is called a critical function if there exist positive constants \(C\) and \(a_0\) so that \[ \rho(y) \leq C \rho(x)\left(1+\frac{\mathrm{d}(x, y)}{\rho(x)}\right)^{\frac{ a_{0}}{a_{0}+1}} \] for all \(x, y \in X\). Definition 1.3. Let \(\rho\) be a critical function on \(X\). For \(p \in (1, \infty)\) and \(\theta \ge 0\), we say that a weight \(w\) belongs to the class \(A_p^{\rho, \theta}\), if \[ [w]_{A_{p}^{\rho, \theta}}=\sup _{B: \text { balls }}\left(\int_{B} w \mathrm{~d} \mu\right)\left(\int_{B} w^{-\frac{1}{p-1}} \mathrm{~d} \mu\right)^{p-1}\left(1+\frac{r_{B}}{\rho\left(x_{B}\right)}\right)^{-\theta} <\infty. \] we say that \(w \in RH_q^{\rho, \theta}\) with \(1<q< \infty\) and \(\theta \ge 0\), if \[ [w]_{RH_{p}^{\rho, \theta}}=\sup _{B: \text { balls }}\left(\int_{B} w^q \mathrm{~d} \mu\right)^{1/q} \left(\int_{B} w \mathrm{~d} \mu\right)^{ -1}\left(1+\frac{r_{B}}{\rho\left(x_{B}\right)}\right)^{-\theta} <\infty. \] The class \(A_p^{\rho, \theta}\) is larger than the Muckenhoupt weights \(A_p\) for \(p\in (1, \infty)\). Assume that \(L\) is a nonnegative self-adjoint operator \(L\) on \(L^2(X)\) and the semigroup \(\mathrm{e}^{-t L}\) has its kernel \(p_t (x, y)\) which satisfies the following estimate: \begin{align*} \left|p_{t}(x, y)\right| \leq \frac{C}{\mu(B(x, \sqrt{t}))} \exp \left(-\frac{\mathrm{d}(x, y)^{2}}{c t}\right)\left(1+\frac{\sqrt{t}}{\rho(x)}+\frac{\sqrt{t}}{\rho(y)}\right)^{-N}\tag{13} \end{align*} for all \(x, y \in X\) and \( t>0\). The authors, by using of the truncated maximal function techniques developed by \textit{A. K. Lerner} and \textit{S. Ombrosi} [New York J. Math. 22, 341--349 (2016; Zbl 1347.42030); J. Geom. Anal. 30, No. 1, 1011--1027 (2020; Zbl 1434.42020)] and a new sparse operator associated to critical functions, obtained the quantitative \(A_p^{\rho, \theta}\) weighted estimate for the spectral multipliers of Laplace transform type for \(L\). The operator defined by \[ \mathfrak{M}(L)=\int_{0}^{\infty} L \mathrm{e}^{-t L} m(t) \mathrm{d} t, \] is bounded on \(L^2(X)\). Theorem 1.5. For \(w\in A_p^{\rho, \theta}\) with \(p\in (1, \infty)\) and \(\theta \ge 0\) we have \[ \|\mathfrak{M}(L) f\|_{L_{w}^{p}(X)} \lesssim[w]_{A_{p}^{\rho, \theta}}^{\max \left\{1, \frac{1}{p-1}\right\}}\|f\|_{L_{w}^{p}(X)}. \] The second main result is a quantitative weighted estimate for the generalized Riesz transform. Assume that \(D\) is a densely defined linear operator on \(L^2(X)\) which possesses the following properties: \par (R1) The generalized Riesz transform \(DL^{-1/2}\) is bounded on \(L ^{p_0}\) for some \(p_0 \in (1, \infty)\). \par (R2) There exits \(\beta > 0\) such that for any \(N > 0\) we have \[ \left[\int_{X}\left|D \mathrm{e}^{-t L}(x, y)\right|^{p_{0}} \mathrm{e}^{\frac{\mathrm{d}(x, y)^{2}}{\beta t}} \mathrm{~d} \mu(x)\right]^{1 / p_{0}} \leq \frac{C_{N}}{\sqrt{t} V(y, \sqrt{t})^{1 / p_{0}^{\prime}}}\left(1+\frac{\sqrt{t}}{\rho(y)}\right)^{-N}. \] Theorem 1.6. Let \(L\) be a nonnegative self-adjoint operator on \(L^2(X)\) satisfying (13) and let \(DL^{-1/2}\) be the generalized Riesz transform satisfying the conditions (R1) and (R2). Then for \(w \in A_{p}^{\rho, \theta_{1}} \cap R H_{\left(p_{0} / p\right)^{\prime}}^{\rho, \theta_{2}}\) with \(p \in (1, p_0)\) and \(\theta_1, \theta_2 \ge 0\), we have \[ \left\|D L^{-1 / 2} f\right\|_{L_{w}^{p}(X)} \lesssim \left([w]_{A_{p}^{\rho, \theta_{1}}}[w]_{R H_{\left(p_{0} / p\right)^{\prime}}^{\rho, \theta_{2}}}\right) ^{\max \left\{\frac{1}{p-\mathrm{I}}, \frac{p_{0}-1}{p_{0}-p}\right\}}\|f\|_{L_{w}^{p}(X)}. \] Theorems 1.5 and 1.6 also give the sharp weighted estimates for the spectral multipliers and the generalized Riesz transforms corresponding to Muckenhoupt weights.