Fractional sobolev regularity of a truncated function

Question

I want to generalize the following result to fractional derivatives, specifically the fractional Laplacian.

Consider a function f which belongs to L2, and all its first order distributional derivatives belong to L2 as well. If I truncate f by subtracting a constant and taking the positive part of the result, then the truncated function is supported on a set of finite measure. Also, its distributional derivatives are all supported on the same finite measure set. By Jensen's or Holder's inequality, I have that f and its distributional derivatives belong to L1 as well as L2.

If I instead have that f and its quarter-Laplacian belong to L2, if I truncate in the same way, will the quarter-Laplacian belong to Lp for some p less than 2? With the extra regularity in the example above, the answer is yes, and can be seen by applying a fractional Gagliardo-Nirenberg interpolation inequality (see http://www.kurims.kyoto-u.ac.jp/~kenkyubu/bessatsu/open/B26/pdf/B26-09.pdf) to interpolate the Lp norm of the fractional Laplacian between the Lp norms of the integer order derivatives.

On a related note, I've seen papers of Adams (http://matwbn.icm.edu.pl/ksiazki/sm/sm89/sm89116.pdf) which show that truncation operators preserve certain functional spaces such as those constructed via Riesz transforms, or Triebel-Lizorkin spaces. However, my question is really about mapping properties of a truncation operator with the range being different than the domain.

Any thoughts appreciated. Thanks!

Answer 1

Let $T_m$ be a standard Fourier multiplier homogeneous with degree $m$ in $\mathbb R^n$ (this the case of your fractional Laplacean). Then $$ \Vert T_m u\Vert_{L^p(\mathbb R^n)}\le C_{m,n}\Vert u\Vert_{W^{s,q}(\mathbb R^n)}, $$ $$ \text{if} \qquad \frac{s-m}{n}=\frac{1}{q}-\frac{1}{p},\quad 1<q<p<+\infty. $$ This is a consequence of Sobolev inequalities and of the continuity of Calderon-Zygmund singular integrals on $L^p$ with $p\in (1,+\infty)$.