As s approaches 0+, s-harmonic functions u_s have asymptotics and s-derivatives expressible via the logarithmic Laplacian of extensions of the exterior data g, yielding pointwise monotonicity in s for many g.
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Improved global and local Gagliardo-Nirenberg inequalities with BMO terms are obtained by using fractional Laplacians in place of local derivatives.
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$s$-harmonic functions in the small order limit
As s approaches 0+, s-harmonic functions u_s have asymptotics and s-derivatives expressible via the logarithmic Laplacian of extensions of the exterior data g, yielding pointwise monotonicity in s for many g.
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Gagliardo-Nirenberg type inequalities with a BMO term and fractional Laplacians
Improved global and local Gagliardo-Nirenberg inequalities with BMO terms are obtained by using fractional Laplacians in place of local derivatives.