Bilinear pseudodifferential operators with symbol in $$BS_{1,1}^m$$ on Triebel–Lizorkin spaces with critical Sobolev index
Collectanea Mathematica - Trang 1-25 - 2023
Tóm tắt
In this paper we obtain new estimates for bilinear pseudodifferential operators with symbol in the class
$$BS_{1,1}^m$$
, when both arguments belong to Triebel-Lizorkin spaces of the type
$$F_{p,q}^{n/p}({\mathbb {R}}^n)$$
. The inequalities are obtained as a consequence of a refinement of the classical Sobolev embedding
$$F^{n/p}_{p,q}({\mathbb {R}}^n)\hookrightarrow \textrm{bmo}({\mathbb {R}}^n)$$
, where we replace
$$\textrm{bmo}({\mathbb {R}}^n)$$
by an appropriate subspace which contains
$$L^\infty ({\mathbb {R}}^n)$$
. As an application, we study the product of functions on
$$F_{p,q}^{n/p}({\mathbb {R}}^n)$$
when
$$1
Tài liệu tham khảo
Arias, S., Rodríguez-López, S.: Some endpoint estimates for bilinear Coifman–Meyer multipliers. J. Math. Anal. Appl. 498(2), 124972 (2021)
Arias, S., Rodríguez-López, S.: Endpoint estimates for bilinear pseudodifferential operators with symbol in \(B S_{1, 1}^m\). J. Math. Anal. Appl. 515(1), 126453 (2022)
Baaske, F., Schmeisser, H.-J.: On the existence and uniqueness of mild and strong solutions of a generalized nonlinear heat equation. Z. Anal. Anwend. 38(3), 287–308 (2019)
Bényi, Á., Nahmod, A.R., Torres, R.H.: Sobolev space estimates and symbolic calculus for bilinear pseudodifferential operators. J. Geom. Anal. 16(no3), 431–453 (2006)
Bényi, Á., Torres, R.H.: Symbolic calculus and the transposes of bilinear pseudodifferential operators. Commun. Partial Differ. Equ. 28(no5–6), 1161–1181 (2003)
Caetano, A.M., Moura, S.D.: Local growth envelopes of spaces of generalised smoothness: the subcritical case. Math. Nachr. 273, 43–57 (2004)
Domínguez, O., Tikhonov, S.: Function spaces of logarithmic smoothness: embeddings and characterizations. Mem. Am. Math. Soc., arXiv preprint arXiv:1811.06399 (2018, to appear)
El-Fallah, O., Kellay, K., Mashreghi, J., Ransford, T.: A primer on the Dirichlet space. Cambridge Tracts in Mathematics, vol. 203. Cambridge University Press, Cambridge (2014)
Feulefack, P.A.: The logarithmic Schrödinger operator and associated Dirichlet problems. J. Math. Anal. Appl. 517(2), 126656 (2023)
Goldberg, D.: A local version of real Hardy spaces. Duke Math. J. 46(no1), 27–42 (1979)
Grafakos, L., Oh, S.: The Kato-Ponce inequality. Commun. Partial Differ. Equ. 39(no6), 1128–1157 (2014)
Grafakos, L., Maldonado, D., Naibo, V.: A remark on an endpoint Kato-Ponce inequality. Differ. Integral Equ. 27(no5–6), 415–424 (2014)
Koezuka, K., Tomita, N.: Bilinear pseudodifferential operators with symbols in \(BS^m_{1,1}\) on Triebel–Lizorkin spaces. J. Fourier Anal. Appl. 24(no1), 309–319 (2018)
Marschall, J.: On the boundedness and compactness of nonregular pseudo-differential operators. Math. Nachr. 175, 231–262 (1995)
Moura, S.: Function spaces of generalised smoothness. Dissert. Math. (Rozprawy Mat.) 398, 88 (2001)
Naibo, V.: On the bilinear Hörmander classes in the scales of Triebel–Lizorkin and Besov spaces. J. Fourier Anal. Appl. 21(no5), 1077–1104 (2015)
Naibo, V.: Bilinear pseudodifferential operators and the Hörmander classes. Not. Am. Math. Soc. 68(no7), 1119–1130 (2021)
Naibo, V., Thomson, A.: Coifman–Meyer multipliers: Leibniz-type rules and applications to scattering of solutions to PDEs. Trans. Am. Math. Soc. 372(no8), 5453–5481 (2019)
Park, B.J.: Equivalence of (quasi-)norms on a vector-valued function space and its applications to multilinear operators. Indiana Univ. Math. J. 70(no5), 1677–1716 (2021)
Runst, T., Sickel, W.: Sobolev spaces of fractional order, Nemytskij operators, and nonlinear partial differential equations. De Gruyter Series in Nonlinear Analysis and Applications, vol. 3. Walter de Gruyter, Berlin (1996)
Rodríguez-López, S., Staubach, W.: Some endpoint estimates for bilinear paraproducts and applications. J. Math. Anal. Appl. 421(2), 1021–1041 (2015)
Sickel, W., Triebel, H.: Hölder inequalities and sharp embeddings in function spaces of \(B^s_{pq}\) and \(F^s_{pq}\) type. Z. Anal. Anwend. 14(1), 105–140 (1995)
Triebel, H.: Theory of Function Spaces. Monographs in Mathematics, vol. 78. Birkhäuser Verlag, Basel (1983)