On Sharp Olsen’s and Trace Inequalities for Multilinear Fractional Integrals
Tóm tắt
We establish a sharp Olsen type inequality
$ \big \| g {\mathcal {I}}_{\alpha }(f_{1}, {\dots } , f_{m}) \big \|_{{L^{q}_{r}} } \leq C \big \| g \big \|_{L^{q}_{\ell } } \prod\limits_{j=1}^{m} \big \| f_{j}\big \|_{L^{p_{j}}_{s_{j}}} $
for multilinear fractional integrals
${\mathcal {I}}_{\alpha }(\vec {f} ) (x) = \int \limits _{({\Bbb {R}}^{n})^{m}}\frac {f_{1}(y_{1}){\cdots } f_{m}(y_{m})}{(|x-y_{1}|+ {\cdots } + |x-y_{m}|)^{mn-\alpha }} d\vec {y}, x\in {\Bbb {R}}^{n}$
, 0 < α < mn, where
${L^{q}_{r}}$
,
$L^{q}_{\ell }$
,
$L^{p_{j}}_{s_{j}}$
, j = 1,…,m, are Morrey space with indices satisfying certain homogeneity conditions. This inequality is sharp because it gives necessary and sufficient condition on a weight function V for which the inequality
$ \big \|{\mathcal {I}}_{\alpha }(f_{1}, {\dots } , f_{m}) \big \|_{{L^{q}_{r}}(V) } \leq C \prod\limits_{j=1}^{m} \big \| f_{j}\big \|_{L^{p_{j}}_{s_{j}}} $
holds. Morrey spaces play an important role in relation to regularity problems of solutions of partial differential equations. They describe the integrability more precisely than Lebesgue spaces. We also derive a characterization of the trace inequality
$ \big \| B_{\alpha } (f_{1},f_{2})\big \|_{{L^{q}_{r}}(d\mu ) } \leq C \prod\limits_{j=1}^{2} \big \| f_{j}\big \|_{L^{p_{j}}_{s_{j}} ({\Bbb {R}}^{n}) }, $
in terms of a Borel measure μ, where Bα is the bilinear fractional integral operator given by the formula
$ B_{\alpha }(f_{1},f_{2})(x) =\int \limits _{{\Bbb {R}}^{n}} \frac {f_{1}(x+t)f_{2}(x-t)}{|t|^{n-\alpha }} dt, 0< \alpha
Tài liệu tham khảo
Adams, D.R.: Traces of potentials arising from translation invariant operators. Ann. Scuola Norm. Sup. Pisa 25, 203–217 (1971)
Adams, D.R.: A trace inequality for generalized potentials. Studia Math. 48, 99–105 (1973)
Adams, D.R.: A note on Riesz potentials, Duke. Math. J. 42, 765–778 (1975)
Chen, X., Xue, Q.: Weighted estimates for a class of multilinear fractional type operators. J. Math. Anal. Appl. 362, 355–373 (2010)
Eridani, A., Kokilashvili, V., Meskhi, A.: Morrey spaces and fractional integral operators. Expo. Math. 27, 227–239 (2009)
Grafakos, L.: On multilinear fractional integrals. Studia Math. 102, 49–56 (1992)
Grafakos, L., Kalton, N.: Some remarks on multilinear maps and interpolation. Math. Ann. 319, 151–180 (2001)
He, Q., Yan, D.: Bilinear fractional integral operators on Morrey spaces. Positivity 25, 399–429 (2021)
Iida, T., Sato, E.: A note on multilinear fractional integrals. Anal. Theory Appl. 26, 301–307 (2010)
Iida, T., Sato, E., Sawano, Y., Tanaka, H.: Weighted norm inequalities for multilinear fractional operators on Morrey spaces. Studia Math. 205, 139–170 (2011)
Iida, T., Sato, E., Sawano, Y., Tanaka, H.: Multilinear fractional integrals on Morrey spaces. Acta Math. Sinica (English Series) 28, 1375–1384 (2012)
Iida, T., Sato, E., Sawano, Y., Tanaka, H.: Sharp bounds for multilinear fractional integral operators on Morrey type spaces. Positivity 16, 339–358 (2012)
Kenig, C., Stein, E.: Multilinear estimates and fractional integration. Math. Res Lett. 6, 1–15 (1999)
Kokilashvili, V., Mastyło, M., Meskhi, A.: On the boundedness of the multilinear fractional integral operators. Nonlin. Anal. Theory Methods Applic. 94, 142–147 (2014)
Kokilashvili, V., Mastyło, M., Meskhi, A.: Two-weight norm estimates for multilinear fractional integrals in classical Lebesgue spaces. Frac. Calc. Appl. Anal. 18, 1146–1163 (2015)
Kokilashvili, V., Mastyło, M., Meskhi, A.: On the boundedness of multilinear fractional integral operators. J. Geom. Anal. 30, 667–679 (2020)
Komori-Furuya, Y.: Weighted estimates for bilinear fractional integral operators: A necessary and sufficient condition for power weights. Collect Math. 71, 25–37 (2020)
Komori, Y., Shirai, S.: Weighted Morrey spaces and a singular integral operator. Math. Nachr. 282, 219–231 (2009)
Li, K., Moen, K., Sun, W.: Sharp weighted inequalities for multilinear fractional maximal operators and fractional integrals. Math. Nachr. 288, 619–632 (2015)
Li, K., Sun, W.: Two weight norm inequalities for the bilinear fractional integrals. Manuscripta Math. 150, 159–175 (2016)
Lerner, A.K., Ombrosi, S., Pérez, C., Torres, R.H., Trujillo–González, R.: New maximal functions and multiple weights for the multilinear Calderón-Zygmund theory. Adv. Math. 220, 1222–1264 (2009)
Maz’ya, V.: Sobolev Spaces. Springer, Berlin and New York (1985)
Moen, K.: Weighted inequalities for multilinear fractional integral operators. Collect. Math. 60, 213–238 (2009)
Moen, K.: New weighted estimates for bilinear fractional integral operators. Trans. Amer. Math. Soc. 366, 627–646 (2014)
Nakamura, S. h.: Generalized weighted Morrey spaces and classical operators. Math. Nachr. 289, 2235–2262 (2016)
Nakamura, S. h., Sawano, Y., Tanaka, H.: The fractional operators on weighted Morrey spaces. J. Geom. Anal. 28, 1502–1524 (2018)
Olsen, P.A.: Fractional integration, Morrey spaces and a Schrödinger equation. Comm. PDE 20, 2005–2055 (1995)
Pan, J., Sun, W.: Two-weight norm inequalities for fractional maximal functions and fractional integral operators on weighted Morrey spaces. Math. Nachr. 293, 970–982 (2020)
Pradolini, G.: Weighted inequalities and pointwise estimates for the multilinear fractional integral and maximal operators. J. Math. Anal. Appl. 367, 640–656 (2010)
Samko, N.: Weighted Hardy and potential operators in Morrey spaces. J. Funct. Spaces Appl., Art. ID 678171, 21 (2012)
Sawano, Y., Sugano, S., Tanaka, H.: Generalized fractional integral operators and fractional maximal operators in the framework of Morrey spaces. Trans. Amer. Math. Soc. 363, 6481–6503 (2011)
Sawano, Y., Sugano, S., Tanaka, H.: Olsen’s inequality and its applications to Schrö,dinger equations. RIMS Kôkyûroku Bessatsu B26, 51–80 (2011)
Shi, Y., Tao, X.: Weighted Lp boundedness for multilinear fractional integral on product spaces, Anal. Theory Appl. 24, 280–291 (2008)
Tang, L.: Endpoint estimates for multilinear fractional integrals. J. Austral. Math. Soc. 84, 419–429 (2008)