We study one-sided nonlocal equations of the form
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on the real line. Notice that to compute this nonlocal operator of order
class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039615007135&_mathId=si2.gif&_user=111111111&_pii=S0022039615007135&_rdoc=1&_issn=00220396&md5=a33b11162d8a9f7234d6b161739b95ad" title="Click to view the MathML source">0<α<1class="mathContainer hidden">class="mathCode"> at a point
class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039615007135&_mathId=si3.gif&_user=111111111&_pii=S0022039615007135&_rdoc=1&_issn=00220396&md5=30a056582f96dfd44ab7c779d2ddfdf3" title="Click to view the MathML source">x0class="mathContainer hidden">class="mathCode"> we need to know the values of
class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039615007135&_mathId=si4.gif&_user=111111111&_pii=S0022039615007135&_rdoc=1&_issn=00220396&md5=31fe1a1816e0a120816c8be0ddadf952" title="Click to view the MathML source">u(x)class="mathContainer hidden">class="mathCode"> to the right of
class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039615007135&_mathId=si3.gif&_user=111111111&_pii=S0022039615007135&_rdoc=1&_issn=00220396&md5=30a056582f96dfd44ab7c779d2ddfdf3" title="Click to view the MathML source">x0class="mathContainer hidden">class="mathCode">, that is, for
class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039615007135&_mathId=si38.gif&_user=111111111&_pii=S0022039615007135&_rdoc=1&_issn=00220396&md5=51ce4e2ed899afab61c4b5262827e01c" title="Click to view the MathML source">x≥x0class="mathContainer hidden">class="mathCode">. We show that the operator above corresponds to a fractional power of a one-sided first order derivative. Maximum principles and a characterization with an extension problem in the spirit of Caffarelli–Silvestre and Stinga–Torrea are proved. It is also shown that these fractional equations can be solved in the general setting of weighted one-sided spaces. In this regard we present suitable inversion results. Along the way we are able to unify and clarify several notions of fractional derivatives found in the literature.