On fractional Orlicz–Sobolev spaces

Alberico, A., Cianchi, A., Pick, L., Slavíková, L.: Fractional Orlicz-Sobolev embeddings. J. Math. Pures Appl., to appear

Alberico, A., Cianchi, A., Pick, L., Slavíková, L.: On the limit as $$s\rightarrow 0^+$$ of fractional Orlicz-Sobolev spaces. J. Fourier Anal. Appl. 26, no. 6, Paper No. 80, 19 pp (2020)

Alberico, A., Cianchi, A., Pick, L., Slavíková, L.: On the limit as $$s\rightarrow 1^-$$ of possibly non-separable fractional Orlicz-Sobolev spaces. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 31, 879–899 (2020)

Alberico, A., Cianchi, A., Pick, L., Slavíková, L.: Compact fractional Orlicz-Sobolev embeddings on domains. Preprint

Almgren, F.J. Jr. Lieb, E.H.: Symmetric decreasing rearrangement is sometimes continuous. J. Amer. Math. Soc. 2 (1989) 683–773

Baernstein II, A.: A unified approach to symmetrization. Partial differential equations of elliptic type (Cortona, : 47–91, p. 1994. Math., XXXV, Cambridge Univ. Press, Cambridge, Sympos (1992)

Barrios, B., Figalli, A., Valdinoci, E.: Bootstrap regularity for integro-differential operators and its application to nonlocal minimal surfaces. Ann. Sc. Norm. Super. Pisa Cl. Sci. 13, 609–639 (2014)

Bellazzini, J., Frank, R., Visciglia, N.: Maximizers for Gagliardo-Nirenberg inequalities and related non-local problems. Math. Ann. 360, 653–673 (2014)

Bourgain, J., Brezis, H., Mironescu, P.: Another look at Sobolev spaces. In: Menaldi, J.L., Rofman, E., Sulem, A. (eds.) Optimal Control and Partial Differential Equations, pp. 439–455. IOS Press, Amsterdam (2001)

Bourgain, J., Brezis, H., Mironescu, P.: Limiting embedding theorems for $$W_{s,p}$$ when $$s\uparrow 1$$ and applications. Dedicated to the memory of Thomas H. Wolff. J. Anal. Math. 87, 77–101 (2002)

Bousquest, P., Ponce, A.C., Van Schaftingen, J.: Strong approximation of fractional Sobolev maps. J. Fixed Point Theory Appl. 15, 133–153 (2014)

Brasco, L., Cinti, E.: On fractional Hardy inequalities in convex sets. Discrete Contin. Dyn. Syst. 38, 4019–4040 (2018)

Brasco, L., Salort, A.: A note on homogeneous Sobolev spaces of fractional order. Ann. Mat. Pura Appl. 198, 1295–1330 (2019)

Brezis, H., Mironescu, P.: Gagliardo-Nirenberg inequalities and non-inequalities: the full story. Ann. Inst. H. Poincaré Anal. Non Linéaire 35, 1255–1376 (2018)

Brezis, H., Mironescu, P.: Where Sobolev interacts with Gagliardo–Nirenberg. J. Funct. Anal. 277, 2839–2864 (2019)

Brezis, H., Nguyen, H.M.: On the distributional Jacobian of maps from $${\mathbb{S}}^{\mathbb{N}}$$ into $${\mathbb{S}}^{\mathbb{N}}$$in fractional Sobolev and Hölder spaces. Ann. of Math. 173, 1141–1183 (2011)

Cabré, X., Fall, M.M., Solà-Morales, J., Weth, T.: Curves and surfaces with constant nonlocal mean curvature: meeting Alexandrov and Delaunay. J. Reine Angew. Math. 745, 253–280 (2018)

Cabré, X., Sire, Y.: Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates. Ann. Inst. H. Poincaré Anal. Non Linéaire 31, 23–53 (2014)

Caffarelli, L., Silvestre, L.: An extension problem related to the fractional Laplacians. Comm. Partial Differ. Equ. 32, 1245–1260 (2007)

Caffarelli, L., Salsa, S., Silvestre, L.: Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian. Invent. Math. 171, 425–461 (2008)

Caffarelli, L., Roquejoffre, S., Savin, O.: Nonlocal minimal surfaces. Commun. Pure Appl. Math. 63, 1111–1144 (2010)

Caffarelli, L., Valdinoci, E.: Uniform estimates and limiting arguments for nonlocal minimal surfaces. Calc. Var. Partial Differ. Equ. 41, 203–240 (2011)

Capolli, M., Maione, A., Salort, A. M., Vecchi, E.: Asymptotic behaviours in Fractional Orlicz-Sobolev spaces on Carnot groups. J. Geom. Anal. 1–34 (2020)

Carrillo, J.A., Figalli, A., Laurent, T., Slepčev, D.: Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations. Duke Math. J. 156, 229–271 (2011)

Chang, S.-Y.A.: Limit of fractional power Sobolev inequalities. J. Funct. Anal. 274, 1177–1201 (2018)

Chen, H., Weth, T.: The Dirichlet problem for the logarithmic Laplacian. Comm. Partial Differ. Equ. 44, 1100–1139 (2019)

Chen, S., Frank, R., Weth, T.: Remainder terms in the fractional Sobolev inequality. Indiana Univ. Math. J. 62, 1381–1397 (2013)

Cianchi, A.: An optimal embedding theorem for Orlicz-Sobolev spaces. Indiana Univ. Math. J. 45, 39–65 (1996)

Cianchi, A.: Boundedness of solutions to variational problems under general growth conditions. Comm. Partial Differ. Equ. 22, 1629–1646 (1997)

Cianchi, A.: Higher-order Sobolev and Poincaré inequalities in Orlicz spaces. Forum Math. 18, 745–767 (2006)

Costa, D.G., de Figueiredo, D.G., Yang, J.: On best constants for limiting embeddings of fractional Sobolev spaces. Adv. Nonlinear Stud. 10, 501–510 (2010)

Dávila, J.: On an open question about functions of bounded variation. Calc. Var. Partial Differ. Equ. 15(4), 519–527 (2002)

De Nápoli, P., Fernández Bonder, J., Salort, A. M.: A Pólya–Szegő principle for general fractional Orlicz–Sobolev spaces. Complex Var. Elliptic Equ. 1–23, (2020)

Dyda, B., Frank, R.: Fractional Hardy–Sobolev–Maz’ya inequality for domains. Studia Math. 208, 151–166 (2012)

Dyda, B., Vähäkangas, A.V.: Characterizations for fractional Hardy inequality. Adv. Calc. Var. 8, 173–182 (2015)

Edmunds, D.E., Gurka, P., Opic, B.: Double exponential integrability of convolution operators in generalized Lorentz–Zygmund spaces. Indiana Univ. Math. J. 44, 19–43 (1995)

Fernández-Bonder, J., Salort, A.M.: Fractional order Orlicz–Sobolev spaces. J. Funct. Anal. 227, 333–367 (2019)

Figalli, A., Fusco, N., Maggi, F., Millot, V., Morini, M.: Isoperimetry and stability properties of balls with respect to nonlocal energies. Commun. Math. Phys. 336, 441–507 (2015)

Filippas, S., Moschini, L., Tertikas, A.: Sharp trace Hardy-Sobolev-Maz’ya inequalities and the fractional Laplacian. Arch. Ration. Mech. Anal. 208, 109–161 (2013)

Frank, R., Jin, T., Xiong. J.: Minimizers for the fractional Sobolev inequality on domains. Calc. Var. Partial Differ. Equ. 57 (2018) Art. 43, 31 pp

Fusco, N., Lions, P.-L., Sbordone, C.: Sobolev imbedding theorems in borderline cases. Proc. Am. Math. Soc. 124, 561–565 (1996)

Garsia, A.M., Rodemich, E.: Monotonicity of certain functionals under rearrangement. Ann. Inst. Fourier 24, 67–116 (1974)

Heuer, N.: On the equivalence of fractional-order Sobolev semi-norms. J. Math. Anal. Appl. 417, 505–518 (2014)

Kuusi, T., Mingione, G., Sire, Y.: Nonlocal self-improving properties. Anal. PDE 8, 57–114 (2015)

Ludwig, M.: Anisotropic fractional Sobolev norms. Adv. Math. 252, 150–157 (2014)

Mallick, A.: Extremals for fractional order Hardy–Sobolev–Maz’ya inequality. Calc. Var. Partial Differ. Equ. 58 (2019) Art. 45, 37 pp

Marano, S., Mosconi, S.J.N.: Asymptotics for optimizers of the fractional Hardy–Sobolev inequality. Commun. Contemp. Math. 21, 1850028 (2019). 33 pp

Maz’ya, V.G.: Sobolev Spaces with Applications to Elliptic Partial Differential Equations. Springer, Berlin (2011)

Maz’ya, V.G., Shaposhnikova, T.: On the Brezis and Mironescu conjecture concerning a Gagliardo–Nirenberg inequality for fractional Sobolev norms. J. Math. Pures Appl. 81, 877–884 (2002)

Maz’ya, V.G., Shaposhnikova, T.: On the Bourgain, Brézis and Mironescu theorem concerning limiting embeddings of fractional Sobolev spaces. J. Funct. Anal. 195, 230–238 (2002)

Maz’ya, V.G., Shaposhnikova, T.: Theory of Sobolev Multipliers, with Applications to Differential and Integral Operators. Springer, Berlin (2009)

Musina, R., Nazarov, A.I.: A note on truncations in fractional Sobolev spaces. Bull. Math. Sci. 9, 1950001 (2019). 7 pp

Palatucci, G., Pisante, A.: Improved Sobolev embeddings, profile decomposition, and concentration-compactness for fractional Sobolev spaces. Calc. Var. Partial Differ. Equ. 50, 799–829 (2014)

Parini, E., Ruf, B.: On the Moser–Trudinger inequality in fractional Sobolev–Slobodeckij spaces. J. Anal. Math. 138, 281–300 (2019)

Pohozhaev, S.I.: On the imbedding Sobolev theorem for $$pl=n$$. Doklady Conference Section Math. Moscow Power Inst. 165, 158–170 (1965). (Russian)

Ros-Oton, X., Serra, J.: The Pohozaev identity for the fractional Laplacian. Arch. Ration. Mech. Anal. 213, 587–628 (2014)

Ros-Oton, X., Serra, J.: Boundary regularity for fully nonlinear integro-differential equations. Duke Math. J. 165, 2079–2154 (2016)

Salort, A.M.: Hardy inequalities in fractional Orlicz-Sobolev spaces. Publ. Mat., to appear

Seeger, A., Trebels, W.: Embeddings for spaces of Lorentz–Sobolev type. Math. Ann. 373, 1017–1056 (2019)

Tzirakis, K.: Sharp trace Hardy–Sobolev inequalities and fractional Hardy–Sobolev inequalities. J. Funct. Anal. 270, 4513–4539 (2016)

Trudinger, N.S.: On imbeddings into Orlicz spaces and some applications. J. Math. Mech. 17, 473–483 (1967)

Van Schaftingen, J., Willem, M.: Set transformations, symmetrizations and isoperimetric inequalities. Nonlinear analysis and applications to physical sciences, 135–152, Springer Italia, Milan, 2004

Yudovich, V.I.: Some estimates connected with integral operators and with solutions of elliptic equations. Soviet Math. Doklady 2, 746–749 (1691)

Zhou, Y.: Fractional Sobolev extension and imbedding. Trans. Am. Math. Soc. 367, 959–979 (2015)